Abstract

For fiber optical sensor made of tapered fiber tip, the effects of the geometrical parameters of tapered tip on two important factors have been investigated. One factor is the intensity of the evanescent wave into fluorescent layer through core-medium interface; the other is the intensity of fluorescence signal transmitted from fluorescent layer to measurement end. A dependence relation of the intensity of fluorescence signal transmitted from fluorescent layer to measurement end upon the geometrical parameters of tapered tip has been obtained. Theoretical results show that the intensity of the evanescent wave into fluorescent layer rises with the decrease of the end diameter of tapered tip, and the increase of the tip length; and the transmitted power of fluorescence signal increases linearly with the increase of the tip length due to the contribution of the side area of tapered tip.

© 2011 OSA

1. Introduction

Fluorescence is a preferred option in biosensing applications by using its information either from the fluorescence intensity or from the fluorescence lifetime. To get a fluorescence signal with a good signal to noise ratio, it is important to use a high intensity excitation light and a high fluorophore concentration together with high collection efficiency from the emitted luminescence [1,2].

On the one hand, for the excitation light, the usual configurations employ the evanescent wave of the guided light through the optical fiber, and the evanescent wave is higher when the excitation light lies down in the UV range [3]. To increase the extension and intensity of the evanescent wave into the fluorescent layer deposited on the optical fiber, tapered optical fibers have been used, for which, the optical fiber is tapered gradually from its maximum diameter down to a constant-diameter waist region of few microns, and then gradually increases back to the original diameter [46]. As the diameter of tapered fiber is less than the core diameter, the light is no longer guided by the core but is guided by the cladding, meanwhile the reduction of fiber core causes the evanescent wave to spread out into the cladding and eventually beyond the outer boundary. By coating extra optical material, for example, a SiO2 or TiO2 sol-gel layer including fluorescent material on the tapered optical fiber [7], the evanescent wave not only interacts with the sensing material through absorption and scattering, but its magnitude is also enhanced in that tapered region.

On the other hand, since the tapering of optical fiber has proved to be a useful method to improve the collection efficiency of fluorescence signal, the tapered optical fiber is provided with the potential to be evanescent wave induced fluorescence based fiber-optic biosensors and has attracted large numbers of attentions. Moreover the theoretical and experimental analysis show the fluorescence collection efficiency is related to the RIs and thicknesses of taper waist and sensing layer [47].

For continuous tapered optical fiber sensor, when the mode fields of the guided light penetrate the cladding deep enough to overlap the sensing material, the effective RIs of guiding light can be modulated by the sensing material [8]. An optimal tapered fiber structure for a short-wavelength-pass filter with sharp and high cutoff efficiency has been achieved and demonstrated [9]. According to three-layer model (core, cladding and external sensing material), the transmission spectra of SMF-28 tapered fibers and the affects of cladding size on the fundamental mode cutoff are theoretically investigated and realized by using numerical finite difference beam propagation method [10]. Especially, B.D. Gupta et al have proposed a theoretical method for surface plasmon resonance based tapered fiber optic sensor, their calculations show that the exponential-linear taper profile with high taper ratio provides the high sensitivity [11].

Another kind of tapered fiber is the tapered fiber tip, which consists of an optical fiber which gradually decreases in diameter until it becomes a tiny tip. Since the tapered tip can transmit the light back to a detector for measurement, the tapered fiber tip coating by fluorescent material is used extensively in biochemical and clinical applications [1215]. Among three kinds of tapered tips (step-etched, conical, and combination tapers), the conical tapered tip has moderate sensitivity since its diameter decreases continuously until it becomes a nanometer radius tip [16].

With the purpose of the miniaturization and in-line monitoring, the conical tapered tip sensor coating by fluorescent material has been investigated in this paper. To make the conical tapered tip sensor have a good fluorescence signal, two important factors must be studied together. One factor is the intensity of the evanescent wave into the fluorescent layer through the core interface; the other is the intensity of the fluorescence signal transmitted from fluorescent layer to fiber core and to the measurement end. Moreover both factors are affected not only by the RIs of fiber core, cladding, fluorescent layer and sample, but also by the geometrical parameters of tapered tip and the thickness of fluorescent layer. In this paper, we place the emphasis on the effects of geometrical parameters of tapered tip upon the intensity of the evanescent wave into the fluorescent layer and the intensity of fluorescence signal transmitted from the fluorescent layer to the measurement end.

2. Excitation Light

For the tapered fiber tip sensor in the wavelength interrogation mode, three-layer model is used to simulate the attenuated total reflection. The first layer is the fiber core, its RI is denoted by n1; the third layer is the sample, whose RI and dielectric constant are denoted by nsam and εsam, respectively. The second layer is the sensing layer with thickness dsen, whose RI and dielectric constant are denoted by nsen and εsen, respectively. According to the Lorentz model of dielectric medium [17], the dielectric constant is given by

εsen(ω)=εsen+Ne2meε0fω02ω2+iωγ,
with
ω0=2πcλmax2,
γ=|Δω0|=2πcΔλmaxλmax2,
and
N=103NAC,
where e = 1.602 × 10−19 C is the elementary charge, me = 9.109 × 10−31 kg is the electron mass, ε0 = 8.85 × 10−12 F/m is the permittivity of vacuum, NA = 6.023 × 1023 mol−1 is the Avagadro’s number, c = 2.998 × 108 m/s is velocity of light in vacuum, εsenis the background dielectric constant, ω0 the absorption frequency, f the oscillator strength, C the molar concentration of absorption oscillators, N the number of absorption oscillators per unit volume, λmax and Δλmax are the absorption maximum wavelength and the full width at half-maximum of absorption spectrum of the sensing layer, respectively.

Figure 1 shows the tapered fiber tip with linear profile. Let a1 (d1) and at (dt) denote the radii (diameters) of fiber core and tip end, respectively, L the tip length, then the tip radius varies with the coordinate z given as

 

Fig. 1 Tapered fiber tip.

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r(z)=a1zL(a1at).

In the tapered region, the angle range of the rays at the coordinate z alters to [ϕ1(z), ϕ2(z)] due to the variation of core diameter, where

ϕ1(z)=cos1a1cosθcrr(z)Ω,
ϕ2(z)=π2Ω,
and the taper angle Ω = tan−1[(a1at)/L], the critical angle θcr = sin−1 (ncl/n1), ncl is the refractive index of the cladding.

As shown in Fig. 1(a), the excitation light from a collimated source is launched into one end of the fiber at the axial point, the normalized transmitted power of p-polarized light arriving at the taper end can be written as [11]

Ptrans=0Ldzϕ1(z)ϕ2(z)RpNref(θ,z)P(θ)dθ0Ldzϕ1(z)ϕ2(z)P(θ)dθ,
where

P(θ)=n12sinθcosθ(1n12cos2θ)2,
Nref(θ,z)=L2ρ(z)tan(θ+Ω).

In above equations, Nref (θ, z) represents the total number of light reflections performed by a ray making angle θ with the normal to the core-cladding layer interface at the coordinate z in the tapered region. The amplitude reflection coefficient for p-polarized incident wave Rp may be calculated according to the multilayer model [11,17]. In this paper, the kind of optical fiber isn’t the emphasis of discussion, so without losing generality, n1 = 1.457, ncl = 1.450 and d1 = 110 μm are fixed.

Firstly, to understand the effect of sensing material, the geometrical parameters of fiber taper dt = 20 μm and L/d1 = 25 are chosen. Let dsen = 500 nm, Δλmax = 100 nm, f = 1.0 be fixed, Fig. 2 shows the transmission spectra of tapered fiber tip sensors (a) with different values of λmax, where C = 0.01 mol/L and εsen = 1.3412 are chosen; (b) with different values of C, where λmax = 600 nm and εsen = 1.3412 are chosen; (c) with different values ofεsen, where λmax = 600 nm and C = 0.01 mol/L are chosen. It can be seen that the resonance wavelength of transmission spectrum is decided by the absorption maximum wavelength, the concentration of absorption oscillators causes greater effect on the strength of transmission spectrum than the background dielectric constant does.

 

Fig. 2 Transmission spectra of tapered fiber tip sensors with (a) different absorption maximum wavelengths, (b) different concentrations of absorption oscillators, and (c) different background dielectric constants. The units are: dsen (nm), λmax (nm), C(mol/L).

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In Fig. 2, the normalized transmitted power Ptrans describes the light power arriving at the tip end, so the difference 1–Ptrans is the power of light escaping from the fiber core into the sensing layer in the tapered region, and may be used to describe the intensity (Peva) of the evanescent wave into the fluorescent layer through the core interface in the tapered region. For the aim of clarity, the lengths of the arrows with solid line denote the maximum intensity of the evanescent wave into the fluorescent layer.

Then, to understand the effects of geometrical parameters of tapered fiber tip, without losing generality, the parameters of sensing layer dsen = 500 nm, λmax = 600 nm, C = 0.01 mol/L, εsen = 1.3412 are fixed. Figure 3(a) and (b) give the transmission spectra and maximum intensities of evanescent wave into fluorescent layer for tapered fiber tip sensors with different values of dt, where L/d1 = 25 is chosen. Figure 4(a) and (b) give the transmission spectra and maximum intensities of evanescent wave into fluorescent layer for tapered fiber tip sensors with different values of L/d1, where dt = 20 μm is chosen. As can be observed that, the intensity of the evanescent wave into fluorescent layer Peva rises with the increase of tip length, but decrease sharply with the increase of tip end diameter.

 

Fig. 3 Transmission spectra (a) and maximum intensity of evanescent wave (b) of tapered fiber tip sensors with different diameters of tip end. The unit of dt is μm.

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Fig. 4 Transmission spectra (a) and maximum intensity of evanescent wave (b) of tapered fiber tip sensors with different tip lengths. The unit of dt is μm.

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3. Fluorescence Signal

The absorptions of excitation photons lead to isotropic emission of fluorescence photons, so the fluorescence signal in sensing layer spread out in all directions. But only parts of fluorescence signal can launch into the core-medium interface, lie within the acceptance cone and arrive at other end of optical fiber.

As shown in Fig. 1(b), let θfi and θft denote the incident and refractive angle of a fluorescence ray, respectively, it is obvious that they obey the law of refraction nsensinθfi=n1sinθft. When the RI of sensing layer nsen is larger than the RI of fiber core, the critical angle of incident fluorescence light is

θfcr=sin1n1nsen.

Only the fluorescence light whose incident angle θfi ∈[0, θfcr] can enter the fiber core, the fluorescence ray whose incident angle θfi ∈[θfcr, π/2] are total internal reflection. As the RI of sensing layer is smaller than the RI of fiber core, the total reflections don’t happen. Progressively, after entering the core-medium interface of tapered tip, only parts of fluorescence whose incident angle θfi ∈[θf1, θf2] can propagate along the fiber and arrive at the measuring end, where

θf1=sin1[n1nsensin(θcrΩ)],
θf2=sin1[n1nsencosΩ].

Supposing the fluorescence material distributes uniformly along the tip surface, and the angular distribution of fluorescence signal is uniform also, then the transmitted power of fluorescence signal arriving at the measuring end can be written as

Pflu=πLd1+dt21+(d1dt2L)2Kθf1θf2Tf(θfi)dθfi,
or
PfluKπd12/4=(1+dtd1)2Ld11+(d1dt2L)2θf1θf2Tf(θfi)dθfi,
where K is a quantity related to the luminous intensity of fluorescence layer, Tf(θfi) is the power transmission coefficient of fluorescence ray at the incident angle θfi, the factor before the dot in Eq. (14) is the area of tip side surface, and the integration behind the dot is called as the integration factor in the following.

By using Fresnel’ formulas, the power transmission coefficient of fluorescence signal from sensing layer to fiber core can be obtained. For s- and p-polarized fluorescence signal, the amplitude transmission coefficients are given as

Es2Es1=2nsencosθfinsencosθfi+n1cosθft,
Ep2Ep1=2nsencosθfin1cosθfi+nsencosθft,
and the power transmission coefficients may be expressed as

Ts(θfi)=n1cosθftnsencosθfi|Es2Es1|2=|4nsenn1cosθficosθft(nsencosθfi+n1cosθft)2|,
Tp(θfi)=n1cosθftnsencosθfi|Ep2Ep1|2=|4nsenn1cosθficosθft(n1cosθfi+nsencosθft)2|.

Both Ts and Tp are the functions of incident angle θfi. In order to obtain the total transmission coefficient Tf for the unpolarized fluorescence, we simply take the mean of Ts and Tp, that is

Tf(θfi)=Ts(θfi)+Tp(θfi)2.

From Eqs. (12)-(20), one can see that, the transmitted power of fluorescence signal Pflu is depended upon the RIs of n1 and nsen, as well as the geometrical parameters of tapered tip: such as tip length L, tip diameters d1 and dt. Since the emphasis of this paper is considering the effects of geometrical parameters on the transmitted power of fluorescence signal, so the RIs n1 = 1.4570 and nsen = 1.3810 are fixed in the following section.

Thirdly, we have investigated the effect of the diameter of tip end dt on the integration factor, where the length L/d1 = 25 is fixed. Figure 5(a) gives the integration values corresponding to different values of dt by open squares. Using the formula Ω = tan−1[(d1dt)/(2L)], we may obtain the corresponding taper angles. Figure 5(b) shows the integration values for different taper angles by open circles. It can be seen that the integration factor increases linearly with the increase of taper angle, and decreases linearly with the increase of the diameter of tip end. Using these calculated data, we can conclude the integration factor may be expressed as

θf1θf2Tf(θfi)dθfi=0.0090+0.220tanΩ  as L/d15,
where
ΩtanΩ=d1dt2L as L/d15,
and the lines in Fig. 5 are drawn according to Eqs. (21) and (22), it can be seen that the formula is well consistent with the calculated data.

 

Fig. 5 Integration factor versus (a) diameter of tip end and (b) taper angle.

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Since the integration factor is dependent upon three quantities n1, nsen and Ω, so the dependence relationship of the transmitted power of fluorescence upon the geometrical parameters of tapered tip can be concluded as

PfluKπd12/4=1+(d1dt2L)2{f1(n1,nsen)(1+dtd1)Ld1+f2(n1,nsen)(1dt2d12)} as Ld15.

For the given RIs of n1 and nsen in this paper, f1(n1, nsen) = 0.0180, f2(n1, nsen) = 0.220. According to Eq. (23), Fig. 6(a) gives the vary curve of the transmitted power of fluorescence with the diameter of tip end.

 

Fig. 6 Transmitted power of fluorescence signal versus diameter of tip end and tip length.

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Finally, we have also considered the effect of tip length on the transmitted power of fluorescence, where the diameter dt = 0 is chosen, so Eq. (23) is simplified as

PfluKπd12/4=1+(d12L)2(0.220+0.0180Ld1).

Figure 6(b) gives the integration values and the transmitted powers of fluorescence corresponding to different tip lengths L/d1 = 2, 3, 4 …100 by open squares and circles, respectively, the solid line is drawn according to Eq. (24), and the dash line is drawn according to Eqs. (21) and (22). The consistency of the calculated data and the fitting equation confirms the validity of Eq. (23).

Therefore, although the integration factor decreases with the increase of length rate L/d1, but the transmitted power of fluorescence increases linearly with the increase of tip length due to the contribution of the area of taper side surface. Since the radical factor closes to 1.0 as L/d1 isn’t smaller than 5, the transmitted power of fluorescence to the measurement end is a quadratic function of the diameters of tapered tip. The quantity K is related to the luminous intensity of fluorescence in the sensing layer, and may be assumed to be in proportion to the intensity of the evanescent wave into fluorescent layer, so one can understand also that the transmitted power of fluorescence is related to the chemical or biological interaction between the fluorescent material and the determinand.

4. Conclusions

In this paper, we have investigated the effects of the geometrical parameters of tapered fiber tip on the intensity of the evanescent wave into fluorescent layer through core-medium interface, and on the intensity of fluorescence signal transmitted from fluorescent layer to the measurement end for the fiber fluorescence biosensor made of tapered fiber tip. The calculation results show that: (1) The intensity of the evanescent wave into fluorescent layer rises with the increase of tip length, but decreases sharply with the increasing diameter of tip end. (2) A dependence relation of the intensity of fluorescence signal transmitted from fluorescent layer to measurement end upon the geometrical parameters of tapered tip has been obtained. The transmitted power of fluorescence signal to the measurement end increases linearly with the length of tapered tip, and varies small with the diameter of tip end. (3) As an optical fluorescence biosensor, optimal geometrical parameters of tapered fiber tip should be that the length rate of tapered tip L/d1≥20 and the diameter rate of tip end dt/d1≤25/110. These results would be useful in designing the fiber optic chemical/biological sensor made of tapered fiber tip.

Acknowledgments

This work is finically supported by the Project of National Science Foundation of China (NSFC) (Number: 50802069).

References and links

1. A. Leung, P. M. Shankar, and R. Mutharasan, “A review of fiber-optic biosensors,” Sens. Actuators B Chem. 125(2), 688–703 (2007). [CrossRef]  

2. C. Pulido and Ó. Esteban, “Improved fluorescence signal with tapered polymer optical fibers under side-illumination,” Sens. Actuators B Chem. 146(1), 190–194 (2010). [CrossRef]  

3. T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005). [CrossRef]   [PubMed]  

4. P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005). [CrossRef]  

5. K. Rijal, A. Leung, P. M. Shankar, and R. Mutharasan, “Detection of pathogen Escherichia coli O157:H7 AT 70 cells/mL using antibody-immobilized biconical tapered fiber sensors,” Biosens. Bioelectron. 21(6), 871–880 (2005). [PubMed]  

6. A. Leung, K. Rijal, P. M. Shankar, and R. Mutharasan, “Effects of geometry on transmission and sensing potential of tapered fiber sensors,” Biosens. Bioelectron. 21(12), 2202–2209 (2006). [CrossRef]   [PubMed]  

7. W. M. Henry, “An investigation of coated tapered optical fibres,” Sens. Actuators B Chem. 22(2), 101–107 (1994). [CrossRef]  

8. N. K. Chen, K. C. Hsu, S. K. Liaw, Y. Lai, and S. Chi, “Influence of depressed-index outer ring on evanescent tunneling loss in tapered double-cladding fibers,” Opt. Lett. 33(15), 1666–1668 (2008). [CrossRef]   [PubMed]  

9. S. Y. Chou, K. C. Hsu, N. K. Chen, S. K. Liaw, Y. S. Chih, Y. Lai, and S. Chi, “Analysis of thermo-optic tunable dispersion-engineered short-wavelength-pass tapered-fiber filters,” J. Lightwave Technol. 27(13), 2208–2215 (2009). [CrossRef]  

10. C. L. Lee, K. H. Lin, and N. K. Chen, “Analysis of optical properties of fundamental-mode in waveguide tapered fibers,” Microelectron. Reliab. 50(5), 726–729 (2010). [CrossRef]  

11. R. K. Verma, A. K. Sharma, and B. D. Gupta, “Surface plasmon resonance based tapered fiber optic sensor with different taper profiles,” Opt. Commun. 281(6), 1486–1491 (2008). [CrossRef]  

12. N. Nath, S. R. Jain, and S. Anand, “Evanescent wave fibre optic sensor for detection of L. donovani specific antibodies in sera of kala azar patients,” Biosens. Bioelectron. 12(6), 491–498 (1997). [CrossRef]   [PubMed]  

13. B. M. Cullum, G. D. Griffin, G. H. Miller, and T. Vo-Dinh, “Intracellular measurements in mammary carcinoma cells using fiber-optic nanosensors,” Anal. Biochem. 277(1), 25–32 (2000). [CrossRef]   [PubMed]  

14. L. Wen-xu and C. Jian, “Continuous monitoring of adriamycin in vivo using fiber optic-based fluorescence chemical sensor,” Anal. Chem. 75(6), 1458–1462 (2003). [CrossRef]   [PubMed]  

15. L. Tang, Y. J. Ren, B. Hong, and K. A. Kang, “Fluorophore-mediated, fiber-optic, multi-analyte, immunosensing system for rapid diagnosis and prognosis of cardiovascular diseases,” J. Biomed. Opt. 11(2), 021011 (2006). [CrossRef]   [PubMed]  

16. G. P. Anderson, J. P. Golden, and F. S. Ligler, “A fiber optic biosensor–combination tapered fibers designed for improved signal acquisition,” Biosens. Bioelectron. 8(5), 249–256 (1993). [CrossRef]  

17. A. K. Sharma and B. D. Gupta, “Absorption-based fiber optic surface plasmon resonance sensor: a theoretical evaluation,” Sens. Actuators B Chem. 100(3), 423–431 (2004). [CrossRef]  

References

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  1. A. Leung, P. M. Shankar, and R. Mutharasan, “A review of fiber-optic biosensors,” Sens. Actuators B Chem. 125(2), 688–703 (2007).
    [CrossRef]
  2. C. Pulido and Ó. Esteban, “Improved fluorescence signal with tapered polymer optical fibers under side-illumination,” Sens. Actuators B Chem. 146(1), 190–194 (2010).
    [CrossRef]
  3. T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
    [CrossRef] [PubMed]
  4. P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005).
    [CrossRef]
  5. K. Rijal, A. Leung, P. M. Shankar, and R. Mutharasan, “Detection of pathogen Escherichia coli O157:H7 AT 70 cells/mL using antibody-immobilized biconical tapered fiber sensors,” Biosens. Bioelectron. 21(6), 871–880 (2005).
    [PubMed]
  6. A. Leung, K. Rijal, P. M. Shankar, and R. Mutharasan, “Effects of geometry on transmission and sensing potential of tapered fiber sensors,” Biosens. Bioelectron. 21(12), 2202–2209 (2006).
    [CrossRef] [PubMed]
  7. W. M. Henry, “An investigation of coated tapered optical fibres,” Sens. Actuators B Chem. 22(2), 101–107 (1994).
    [CrossRef]
  8. N. K. Chen, K. C. Hsu, S. K. Liaw, Y. Lai, and S. Chi, “Influence of depressed-index outer ring on evanescent tunneling loss in tapered double-cladding fibers,” Opt. Lett. 33(15), 1666–1668 (2008).
    [CrossRef] [PubMed]
  9. S. Y. Chou, K. C. Hsu, N. K. Chen, S. K. Liaw, Y. S. Chih, Y. Lai, and S. Chi, “Analysis of thermo-optic tunable dispersion-engineered short-wavelength-pass tapered-fiber filters,” J. Lightwave Technol. 27(13), 2208–2215 (2009).
    [CrossRef]
  10. C. L. Lee, K. H. Lin, and N. K. Chen, “Analysis of optical properties of fundamental-mode in waveguide tapered fibers,” Microelectron. Reliab. 50(5), 726–729 (2010).
    [CrossRef]
  11. R. K. Verma, A. K. Sharma, and B. D. Gupta, “Surface plasmon resonance based tapered fiber optic sensor with different taper profiles,” Opt. Commun. 281(6), 1486–1491 (2008).
    [CrossRef]
  12. N. Nath, S. R. Jain, and S. Anand, “Evanescent wave fibre optic sensor for detection of L. donovani specific antibodies in sera of kala azar patients,” Biosens. Bioelectron. 12(6), 491–498 (1997).
    [CrossRef] [PubMed]
  13. B. M. Cullum, G. D. Griffin, G. H. Miller, and T. Vo-Dinh, “Intracellular measurements in mammary carcinoma cells using fiber-optic nanosensors,” Anal. Biochem. 277(1), 25–32 (2000).
    [CrossRef] [PubMed]
  14. L. Wen-xu and C. Jian, “Continuous monitoring of adriamycin in vivo using fiber optic-based fluorescence chemical sensor,” Anal. Chem. 75(6), 1458–1462 (2003).
    [CrossRef] [PubMed]
  15. L. Tang, Y. J. Ren, B. Hong, and K. A. Kang, “Fluorophore-mediated, fiber-optic, multi-analyte, immunosensing system for rapid diagnosis and prognosis of cardiovascular diseases,” J. Biomed. Opt. 11(2), 021011 (2006).
    [CrossRef] [PubMed]
  16. G. P. Anderson, J. P. Golden, and F. S. Ligler, “A fiber optic biosensor–combination tapered fibers designed for improved signal acquisition,” Biosens. Bioelectron. 8(5), 249–256 (1993).
    [CrossRef]
  17. A. K. Sharma and B. D. Gupta, “Absorption-based fiber optic surface plasmon resonance sensor: a theoretical evaluation,” Sens. Actuators B Chem. 100(3), 423–431 (2004).
    [CrossRef]

2010 (2)

C. Pulido and Ó. Esteban, “Improved fluorescence signal with tapered polymer optical fibers under side-illumination,” Sens. Actuators B Chem. 146(1), 190–194 (2010).
[CrossRef]

C. L. Lee, K. H. Lin, and N. K. Chen, “Analysis of optical properties of fundamental-mode in waveguide tapered fibers,” Microelectron. Reliab. 50(5), 726–729 (2010).
[CrossRef]

2009 (1)

2008 (2)

N. K. Chen, K. C. Hsu, S. K. Liaw, Y. Lai, and S. Chi, “Influence of depressed-index outer ring on evanescent tunneling loss in tapered double-cladding fibers,” Opt. Lett. 33(15), 1666–1668 (2008).
[CrossRef] [PubMed]

R. K. Verma, A. K. Sharma, and B. D. Gupta, “Surface plasmon resonance based tapered fiber optic sensor with different taper profiles,” Opt. Commun. 281(6), 1486–1491 (2008).
[CrossRef]

2007 (1)

A. Leung, P. M. Shankar, and R. Mutharasan, “A review of fiber-optic biosensors,” Sens. Actuators B Chem. 125(2), 688–703 (2007).
[CrossRef]

2006 (2)

L. Tang, Y. J. Ren, B. Hong, and K. A. Kang, “Fluorophore-mediated, fiber-optic, multi-analyte, immunosensing system for rapid diagnosis and prognosis of cardiovascular diseases,” J. Biomed. Opt. 11(2), 021011 (2006).
[CrossRef] [PubMed]

A. Leung, K. Rijal, P. M. Shankar, and R. Mutharasan, “Effects of geometry on transmission and sensing potential of tapered fiber sensors,” Biosens. Bioelectron. 21(12), 2202–2209 (2006).
[CrossRef] [PubMed]

2005 (3)

T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
[CrossRef] [PubMed]

P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005).
[CrossRef]

K. Rijal, A. Leung, P. M. Shankar, and R. Mutharasan, “Detection of pathogen Escherichia coli O157:H7 AT 70 cells/mL using antibody-immobilized biconical tapered fiber sensors,” Biosens. Bioelectron. 21(6), 871–880 (2005).
[PubMed]

2004 (1)

A. K. Sharma and B. D. Gupta, “Absorption-based fiber optic surface plasmon resonance sensor: a theoretical evaluation,” Sens. Actuators B Chem. 100(3), 423–431 (2004).
[CrossRef]

2003 (1)

L. Wen-xu and C. Jian, “Continuous monitoring of adriamycin in vivo using fiber optic-based fluorescence chemical sensor,” Anal. Chem. 75(6), 1458–1462 (2003).
[CrossRef] [PubMed]

2000 (1)

B. M. Cullum, G. D. Griffin, G. H. Miller, and T. Vo-Dinh, “Intracellular measurements in mammary carcinoma cells using fiber-optic nanosensors,” Anal. Biochem. 277(1), 25–32 (2000).
[CrossRef] [PubMed]

1997 (1)

N. Nath, S. R. Jain, and S. Anand, “Evanescent wave fibre optic sensor for detection of L. donovani specific antibodies in sera of kala azar patients,” Biosens. Bioelectron. 12(6), 491–498 (1997).
[CrossRef] [PubMed]

1994 (1)

W. M. Henry, “An investigation of coated tapered optical fibres,” Sens. Actuators B Chem. 22(2), 101–107 (1994).
[CrossRef]

1993 (1)

G. P. Anderson, J. P. Golden, and F. S. Ligler, “A fiber optic biosensor–combination tapered fibers designed for improved signal acquisition,” Biosens. Bioelectron. 8(5), 249–256 (1993).
[CrossRef]

Anand, S.

N. Nath, S. R. Jain, and S. Anand, “Evanescent wave fibre optic sensor for detection of L. donovani specific antibodies in sera of kala azar patients,” Biosens. Bioelectron. 12(6), 491–498 (1997).
[CrossRef] [PubMed]

Anderson, G. P.

G. P. Anderson, J. P. Golden, and F. S. Ligler, “A fiber optic biosensor–combination tapered fibers designed for improved signal acquisition,” Biosens. Bioelectron. 8(5), 249–256 (1993).
[CrossRef]

Battaglia, T. M.

T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
[CrossRef] [PubMed]

Beaudoin, S. P.

T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
[CrossRef] [PubMed]

Booksh, K. S.

T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
[CrossRef] [PubMed]

Caldas, P.

P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005).
[CrossRef]

Chen, N. K.

Chi, S.

Chih, Y. S.

Chou, S. Y.

Cullum, B. M.

B. M. Cullum, G. D. Griffin, G. H. Miller, and T. Vo-Dinh, “Intracellular measurements in mammary carcinoma cells using fiber-optic nanosensors,” Anal. Biochem. 277(1), 25–32 (2000).
[CrossRef] [PubMed]

Esteban, Ó.

C. Pulido and Ó. Esteban, “Improved fluorescence signal with tapered polymer optical fibers under side-illumination,” Sens. Actuators B Chem. 146(1), 190–194 (2010).
[CrossRef]

Farahi, F.

P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005).
[CrossRef]

Foster, K. N.

T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
[CrossRef] [PubMed]

Golden, J. P.

G. P. Anderson, J. P. Golden, and F. S. Ligler, “A fiber optic biosensor–combination tapered fibers designed for improved signal acquisition,” Biosens. Bioelectron. 8(5), 249–256 (1993).
[CrossRef]

Griffin, G. D.

B. M. Cullum, G. D. Griffin, G. H. Miller, and T. Vo-Dinh, “Intracellular measurements in mammary carcinoma cells using fiber-optic nanosensors,” Anal. Biochem. 277(1), 25–32 (2000).
[CrossRef] [PubMed]

Gupta, B. D.

R. K. Verma, A. K. Sharma, and B. D. Gupta, “Surface plasmon resonance based tapered fiber optic sensor with different taper profiles,” Opt. Commun. 281(6), 1486–1491 (2008).
[CrossRef]

A. K. Sharma and B. D. Gupta, “Absorption-based fiber optic surface plasmon resonance sensor: a theoretical evaluation,” Sens. Actuators B Chem. 100(3), 423–431 (2004).
[CrossRef]

Henry, W. M.

W. M. Henry, “An investigation of coated tapered optical fibres,” Sens. Actuators B Chem. 22(2), 101–107 (1994).
[CrossRef]

Holloway, G. A.

T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
[CrossRef] [PubMed]

Hong, B.

L. Tang, Y. J. Ren, B. Hong, and K. A. Kang, “Fluorophore-mediated, fiber-optic, multi-analyte, immunosensing system for rapid diagnosis and prognosis of cardiovascular diseases,” J. Biomed. Opt. 11(2), 021011 (2006).
[CrossRef] [PubMed]

Hsu, K. C.

Jain, S. R.

N. Nath, S. R. Jain, and S. Anand, “Evanescent wave fibre optic sensor for detection of L. donovani specific antibodies in sera of kala azar patients,” Biosens. Bioelectron. 12(6), 491–498 (1997).
[CrossRef] [PubMed]

Jian, C.

L. Wen-xu and C. Jian, “Continuous monitoring of adriamycin in vivo using fiber optic-based fluorescence chemical sensor,” Anal. Chem. 75(6), 1458–1462 (2003).
[CrossRef] [PubMed]

Jorge, P. A. S.

P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005).
[CrossRef]

Kang, K. A.

L. Tang, Y. J. Ren, B. Hong, and K. A. Kang, “Fluorophore-mediated, fiber-optic, multi-analyte, immunosensing system for rapid diagnosis and prognosis of cardiovascular diseases,” J. Biomed. Opt. 11(2), 021011 (2006).
[CrossRef] [PubMed]

Lai, Y.

Lee, C. L.

C. L. Lee, K. H. Lin, and N. K. Chen, “Analysis of optical properties of fundamental-mode in waveguide tapered fibers,” Microelectron. Reliab. 50(5), 726–729 (2010).
[CrossRef]

Leung, A.

A. Leung, P. M. Shankar, and R. Mutharasan, “A review of fiber-optic biosensors,” Sens. Actuators B Chem. 125(2), 688–703 (2007).
[CrossRef]

A. Leung, K. Rijal, P. M. Shankar, and R. Mutharasan, “Effects of geometry on transmission and sensing potential of tapered fiber sensors,” Biosens. Bioelectron. 21(12), 2202–2209 (2006).
[CrossRef] [PubMed]

K. Rijal, A. Leung, P. M. Shankar, and R. Mutharasan, “Detection of pathogen Escherichia coli O157:H7 AT 70 cells/mL using antibody-immobilized biconical tapered fiber sensors,” Biosens. Bioelectron. 21(6), 871–880 (2005).
[PubMed]

Liaw, S. K.

Ligler, F. S.

G. P. Anderson, J. P. Golden, and F. S. Ligler, “A fiber optic biosensor–combination tapered fibers designed for improved signal acquisition,” Biosens. Bioelectron. 8(5), 249–256 (1993).
[CrossRef]

Lin, K. H.

C. L. Lee, K. H. Lin, and N. K. Chen, “Analysis of optical properties of fundamental-mode in waveguide tapered fibers,” Microelectron. Reliab. 50(5), 726–729 (2010).
[CrossRef]

Masson, J. F.

T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
[CrossRef] [PubMed]

Miller, G. H.

B. M. Cullum, G. D. Griffin, G. H. Miller, and T. Vo-Dinh, “Intracellular measurements in mammary carcinoma cells using fiber-optic nanosensors,” Anal. Biochem. 277(1), 25–32 (2000).
[CrossRef] [PubMed]

Mutharasan, R.

A. Leung, P. M. Shankar, and R. Mutharasan, “A review of fiber-optic biosensors,” Sens. Actuators B Chem. 125(2), 688–703 (2007).
[CrossRef]

A. Leung, K. Rijal, P. M. Shankar, and R. Mutharasan, “Effects of geometry on transmission and sensing potential of tapered fiber sensors,” Biosens. Bioelectron. 21(12), 2202–2209 (2006).
[CrossRef] [PubMed]

K. Rijal, A. Leung, P. M. Shankar, and R. Mutharasan, “Detection of pathogen Escherichia coli O157:H7 AT 70 cells/mL using antibody-immobilized biconical tapered fiber sensors,” Biosens. Bioelectron. 21(6), 871–880 (2005).
[PubMed]

Nath, N.

N. Nath, S. R. Jain, and S. Anand, “Evanescent wave fibre optic sensor for detection of L. donovani specific antibodies in sera of kala azar patients,” Biosens. Bioelectron. 12(6), 491–498 (1997).
[CrossRef] [PubMed]

Oliva, A. G.

P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005).
[CrossRef]

Pulido, C.

C. Pulido and Ó. Esteban, “Improved fluorescence signal with tapered polymer optical fibers under side-illumination,” Sens. Actuators B Chem. 146(1), 190–194 (2010).
[CrossRef]

Ren, Y. J.

L. Tang, Y. J. Ren, B. Hong, and K. A. Kang, “Fluorophore-mediated, fiber-optic, multi-analyte, immunosensing system for rapid diagnosis and prognosis of cardiovascular diseases,” J. Biomed. Opt. 11(2), 021011 (2006).
[CrossRef] [PubMed]

Rijal, K.

A. Leung, K. Rijal, P. M. Shankar, and R. Mutharasan, “Effects of geometry on transmission and sensing potential of tapered fiber sensors,” Biosens. Bioelectron. 21(12), 2202–2209 (2006).
[CrossRef] [PubMed]

K. Rijal, A. Leung, P. M. Shankar, and R. Mutharasan, “Detection of pathogen Escherichia coli O157:H7 AT 70 cells/mL using antibody-immobilized biconical tapered fiber sensors,” Biosens. Bioelectron. 21(6), 871–880 (2005).
[PubMed]

Rogers, J.

T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
[CrossRef] [PubMed]

Rosa, C. C.

P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005).
[CrossRef]

Santos, J. L.

P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005).
[CrossRef]

Shankar, P. M.

A. Leung, P. M. Shankar, and R. Mutharasan, “A review of fiber-optic biosensors,” Sens. Actuators B Chem. 125(2), 688–703 (2007).
[CrossRef]

A. Leung, K. Rijal, P. M. Shankar, and R. Mutharasan, “Effects of geometry on transmission and sensing potential of tapered fiber sensors,” Biosens. Bioelectron. 21(12), 2202–2209 (2006).
[CrossRef] [PubMed]

K. Rijal, A. Leung, P. M. Shankar, and R. Mutharasan, “Detection of pathogen Escherichia coli O157:H7 AT 70 cells/mL using antibody-immobilized biconical tapered fiber sensors,” Biosens. Bioelectron. 21(6), 871–880 (2005).
[PubMed]

Sharma, A. K.

R. K. Verma, A. K. Sharma, and B. D. Gupta, “Surface plasmon resonance based tapered fiber optic sensor with different taper profiles,” Opt. Commun. 281(6), 1486–1491 (2008).
[CrossRef]

A. K. Sharma and B. D. Gupta, “Absorption-based fiber optic surface plasmon resonance sensor: a theoretical evaluation,” Sens. Actuators B Chem. 100(3), 423–431 (2004).
[CrossRef]

Sierks, M. R.

T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
[CrossRef] [PubMed]

Silva, J. C. G. E. D.

P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005).
[CrossRef]

Tang, L.

L. Tang, Y. J. Ren, B. Hong, and K. A. Kang, “Fluorophore-mediated, fiber-optic, multi-analyte, immunosensing system for rapid diagnosis and prognosis of cardiovascular diseases,” J. Biomed. Opt. 11(2), 021011 (2006).
[CrossRef] [PubMed]

Verma, R. K.

R. K. Verma, A. K. Sharma, and B. D. Gupta, “Surface plasmon resonance based tapered fiber optic sensor with different taper profiles,” Opt. Commun. 281(6), 1486–1491 (2008).
[CrossRef]

Vo-Dinh, T.

B. M. Cullum, G. D. Griffin, G. H. Miller, and T. Vo-Dinh, “Intracellular measurements in mammary carcinoma cells using fiber-optic nanosensors,” Anal. Biochem. 277(1), 25–32 (2000).
[CrossRef] [PubMed]

Wen-xu, L.

L. Wen-xu and C. Jian, “Continuous monitoring of adriamycin in vivo using fiber optic-based fluorescence chemical sensor,” Anal. Chem. 75(6), 1458–1462 (2003).
[CrossRef] [PubMed]

Anal. Biochem. (1)

B. M. Cullum, G. D. Griffin, G. H. Miller, and T. Vo-Dinh, “Intracellular measurements in mammary carcinoma cells using fiber-optic nanosensors,” Anal. Biochem. 277(1), 25–32 (2000).
[CrossRef] [PubMed]

Anal. Chem. (2)

L. Wen-xu and C. Jian, “Continuous monitoring of adriamycin in vivo using fiber optic-based fluorescence chemical sensor,” Anal. Chem. 75(6), 1458–1462 (2003).
[CrossRef] [PubMed]

T. M. Battaglia, J. F. Masson, M. R. Sierks, S. P. Beaudoin, J. Rogers, K. N. Foster, G. A. Holloway, and K. S. Booksh, “Quantification of cytokines involved in wound healing using surface plasmon resonance,” Anal. Chem. 77(21), 7016–7023 (2005).
[CrossRef] [PubMed]

Biosens. Bioelectron. (4)

K. Rijal, A. Leung, P. M. Shankar, and R. Mutharasan, “Detection of pathogen Escherichia coli O157:H7 AT 70 cells/mL using antibody-immobilized biconical tapered fiber sensors,” Biosens. Bioelectron. 21(6), 871–880 (2005).
[PubMed]

A. Leung, K. Rijal, P. M. Shankar, and R. Mutharasan, “Effects of geometry on transmission and sensing potential of tapered fiber sensors,” Biosens. Bioelectron. 21(12), 2202–2209 (2006).
[CrossRef] [PubMed]

G. P. Anderson, J. P. Golden, and F. S. Ligler, “A fiber optic biosensor–combination tapered fibers designed for improved signal acquisition,” Biosens. Bioelectron. 8(5), 249–256 (1993).
[CrossRef]

N. Nath, S. R. Jain, and S. Anand, “Evanescent wave fibre optic sensor for detection of L. donovani specific antibodies in sera of kala azar patients,” Biosens. Bioelectron. 12(6), 491–498 (1997).
[CrossRef] [PubMed]

Fiber Integr. Opt. (1)

P. A. S. Jorge, P. Caldas, J. C. G. E. D. Silva, C. C. Rosa, A. G. Oliva, J. L. Santos, and F. Farahi, “Luminiscence-based optical fiber chemical sensors,” Fiber Integr. Opt. 24(3), 201–225 (2005).
[CrossRef]

J. Biomed. Opt. (1)

L. Tang, Y. J. Ren, B. Hong, and K. A. Kang, “Fluorophore-mediated, fiber-optic, multi-analyte, immunosensing system for rapid diagnosis and prognosis of cardiovascular diseases,” J. Biomed. Opt. 11(2), 021011 (2006).
[CrossRef] [PubMed]

J. Lightwave Technol. (1)

Microelectron. Reliab. (1)

C. L. Lee, K. H. Lin, and N. K. Chen, “Analysis of optical properties of fundamental-mode in waveguide tapered fibers,” Microelectron. Reliab. 50(5), 726–729 (2010).
[CrossRef]

Opt. Commun. (1)

R. K. Verma, A. K. Sharma, and B. D. Gupta, “Surface plasmon resonance based tapered fiber optic sensor with different taper profiles,” Opt. Commun. 281(6), 1486–1491 (2008).
[CrossRef]

Opt. Lett. (1)

Sens. Actuators B Chem. (4)

A. K. Sharma and B. D. Gupta, “Absorption-based fiber optic surface plasmon resonance sensor: a theoretical evaluation,” Sens. Actuators B Chem. 100(3), 423–431 (2004).
[CrossRef]

W. M. Henry, “An investigation of coated tapered optical fibres,” Sens. Actuators B Chem. 22(2), 101–107 (1994).
[CrossRef]

A. Leung, P. M. Shankar, and R. Mutharasan, “A review of fiber-optic biosensors,” Sens. Actuators B Chem. 125(2), 688–703 (2007).
[CrossRef]

C. Pulido and Ó. Esteban, “Improved fluorescence signal with tapered polymer optical fibers under side-illumination,” Sens. Actuators B Chem. 146(1), 190–194 (2010).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Tapered fiber tip.

Fig. 2
Fig. 2

Transmission spectra of tapered fiber tip sensors with (a) different absorption maximum wavelengths, (b) different concentrations of absorption oscillators, and (c) different background dielectric constants. The units are: dsen (nm), λmax (nm), C(mol/L).

Fig. 3
Fig. 3

Transmission spectra (a) and maximum intensity of evanescent wave (b) of tapered fiber tip sensors with different diameters of tip end. The unit of dt is μm.

Fig. 4
Fig. 4

Transmission spectra (a) and maximum intensity of evanescent wave (b) of tapered fiber tip sensors with different tip lengths. The unit of dt is μm.

Fig. 5
Fig. 5

Integration factor versus (a) diameter of tip end and (b) taper angle.

Fig. 6
Fig. 6

Transmitted power of fluorescence signal versus diameter of tip end and tip length.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

ε sen (ω)= ε sen + N e 2 m e ε 0 f ω 0 2 ω 2 +iωγ ,
ω 0 = 2πc λ max 2 ,
γ=|Δ ω 0 |=2πc Δ λ max λ max 2 ,
N= 10 3 N A C,
r(z)= a 1 z L ( a 1 a t ).
ϕ 1 (z)= cos 1 a 1 cos θ cr r(z) Ω,
ϕ 2 (z)= π 2 Ω,
P trans = 0 L dz ϕ 1 (z) ϕ 2 (z) R p N ref (θ,z) P(θ)dθ 0 L dz ϕ 1 (z) ϕ 2 (z) P(θ)dθ ,
P(θ)= n 1 2 sinθcosθ (1 n 1 2 cos 2 θ) 2 ,
N ref (θ,z)= L 2ρ(z)tan(θ+Ω) .
θ fcr = sin 1 n 1 n sen .
θ f1 = sin 1 [ n 1 n sen sin( θ cr Ω)],
θ f2 = sin 1 [ n 1 n sen cosΩ].
P flu =πL d 1 + d t 2 1+ ( d 1 d t 2L ) 2 K θ f1 θ f2 T f ( θ fi ) d θ fi ,
P flu Kπ d 1 2 /4 =(1+ d t d 1 ) 2L d 1 1+ ( d 1 d t 2L ) 2 θ f1 θ f2 T f ( θ fi ) d θ fi ,
E s2 E s1 = 2 n sen cos θ fi n sen cos θ fi + n 1 cos θ ft ,
E p2 E p1 = 2 n sen cos θ fi n 1 cos θ fi + n sen cos θ ft ,
T s ( θ fi )= n 1 cos θ ft n sen cos θ fi | E s2 E s1 | 2 =| 4 n sen n 1 cos θ fi cos θ ft ( n sen cos θ fi + n 1 cos θ ft ) 2 |,
T p ( θ fi )= n 1 cos θ ft n sen cos θ fi | E p2 E p1 | 2 =| 4 n sen n 1 cos θ fi cos θ ft ( n 1 cos θ fi + n sen cos θ ft ) 2 |.
T f ( θ fi )= T s ( θ fi )+ T p ( θ fi ) 2 .
θ f1 θ f2 T f ( θ fi ) d θ fi =0.0090+0.220tanΩ  as L/ d 1 5,
ΩtanΩ= d 1 d t 2L  as L/ d 1 5,
P flu Kπ d 1 2 /4 = 1+ ( d 1 d t 2L ) 2 { f 1 ( n 1 , n sen )(1+ d t d 1 ) L d 1 + f 2 ( n 1 , n sen )(1 d t 2 d 1 2 ) } as  L d 1 5.
P flu Kπ d 1 2 /4 = 1+ ( d 1 2L ) 2 (0.220+0.0180 L d 1 ).

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