Abstract

We numerically investigate a D-shaped fiber surface plasmon resonance sensor based on all-solid photonic crystal fiber (PCF) with finite element method. In the side-polished PCF sensor, field leakage is guided to penetrate through the gap between the rods, causing a pronounced phase modulation in the deep polishing case. Taking advantage of these amplified phase shifts, a high-performance fiber sensor design is proposed. The significant enhancements arising from this new sensor design should lift the performance of the fiber SPR sensor into the range capable of detecting a wide range of biochemical interactions, which makes it especially attractive for many in vivo and in situ bioanalysis applications. Several parameters which influence the field leakage, such as the polishing position, the pitch of the PCF, and the rod diameter, are inspected to evaluate their impacts. Furthermore, we develop a mathematical model to describe the effects of varying the structural parameters of a D-shaped PCF sensor on the evanescent field and the sensor performance.

© 2014 Optical Society of America

1. Introduction

The surface plasmon resonance (SPR) technique has become popular in biochemistry over the last decade for its reliability and precision in quantifying binding processes [1]. The fiber SPR sensor was first demonstrated by Jorgenson et al. in 1993 [2], by integrating SPR on the cylindrical surface of an optical fiber. This extremely compact device is unique among the developing SPR techniques, especially with regard to its potential applications in carrying out in vivo and in situ measurements [3]. Some groups have used fiber SPR sensors for DNA research and cell analysis, where the large mass load on the surface is sufficient to produce a wavelength shift [4, 5]. However, due to the high modal noise, it is difficult for conventional multi-mode fiber SPR sensor to detect small biomolecules and molecular compounds, which is of great interest in biochemistry research. Application development calls for a new fiber SPR sensor design. In general, there are two ways to stimulate a plasmonic wave for fiber sensing: by setting the interface inside the fiber [6, 7], in particular, by integrating a microfluidic channel into the fiber [810]; or, by leaking the electromagnetic field out, exemplified by the nanofiber [11, 12], grating, and side polishing methods. Of these approaches, microfluidic integration suffers from difficulties with film deposition and interim verification, while nanofibers are too fragile for a practical device. More viable is the grating approach (e.g. tilted fiber Bragg grating), in which cladding modes are generated by the period structure, propagating along the surface and stimulating plasmonic waves on the interface. Fiber-optic biosensors based on tilted fiber Bragg gratings have been demonstrated experimentally with high sensitivity and low limit of detection [1316]. Another promising choice is D-shaped fibers [1722], which are commonly produced by mechanical polishing and chemical etching of base fibers.

As has been reported in previous studies, phase modulation is essential in SPR phenomena, and phase interrogation is the state-of-the-art in SPR sensing. Kabashin et al. report that phase interrogation can provide at least a two-orders-of-magnitude improvement in the detection limit compared to an amplitude-based scheme, due to its physical characteristics and low-noise design [23].

In [24] the authors describe an ultra-high sensitivity polarimetric sensor based on SPR for strain measurement via the change of the refractive index of the fiber core. They use a multimode fiber with a diameter of 70 μm, which is side-polished to form a four-layer D-shaped structure. Taking advantage of the birefringence of the D-shaped structure, an interferometry is constructed for phase demodulation. Since the s-wave is independent of the resonance, the SPR phase change within the p-wave can be extracted by the demodulation process. This experimental setup can serve as a paradigm for the claim of Kabashin. However, the output performance is unsatisfactory — in particular, the baseline noise level is 0.4 Deg, which is much higher than that reported for a prism SPR (0.01 Deg) [25]. In addition, the resonance was intensified by immersing the sensor in a high refractive index solution of ne = 1.44, and this effect should be much weaker under water. We attribute these weaknesses to the sensor design, where the multimode fiber should counter for the noise level and the weak resonance. An alternative scheme may overcome these problems.

In [22] a D-shaped all-solid photonic crystal fiber (PCF) sensor based on SPR was proposed. Since all-solid PCF is mechanically uniform, it is regarded as an ideal substrate for D-shaped sensor fabrication. However, for sensing protocols based on wavelength interrogation, the performance of the proposed fiber sensor is limited, only comparable to that of sensors based on conventional multi-mode fibers. Thus we are interested in its phase performance, especially for its structured cross section, which should introduce a difference in the field leakage.

In this paper, we use numerical simulations to investigate side-polished D-shaped fiber SPR sensors based on all-solid PCFs, and identify its phase shift; in particular, we describe a pronounced wave vector distortion for deep polishing and analyze the phase modulation around the resonance wavelength. Based on these findings, we present a design for a phase-sensitive fiber SPR sensor system based on a D-shaped all-solid photonic crystal fiber, and assess the dependence of performance on the sensor configuration.

2. Schematic and method

The base fiber is a five-layer all-solid PCF with uniform rod radius. An enlarged schematic of the fiber sensor is shown in the upper half of Fig. 1. The central zone in red is the core region while the surrounding low-index rods make up the cladding. The lattice pitch is Λ = 2 μm, and the radius of the low-index rods is 0.4 Λ. The refractive indices of the core and cladding rods are 1.46 and 1.44, respectively. The fiber is side-polished to form a flat plane where the gold film is deposited. The gold film has a fixed thickness t = 45 nm. The complex relative permittivity of the gold is characterized by the Drude-Lorentz model with one additional Lorentzian term [26]. We denote the environmental refractive index as ne, which represents the analyte in biochemical applications. Changes to the mass load on the polished plane – for example, changes associated with the progress of antibody-antigen binding – will change the refractive index perceived by the evanescent field. In our simulations, the primary variable is the polishing depth, measured as the height h of the flat surface above the center of the fiber. By controlling the height of the surface resulting from the polishing process, we can directly tune the intensity of the field leakage. We mark the gap between the rods in the fiber core region and the metal film as the leakage channel, as shown in Fig. 1. The analysis of the electric field density along this channel will help to model the relationship between the evanescent field distribution and the sensor performance.

 figure: Fig. 1

Fig. 1 Enlarged schematic of fiber SPR sensor based on an all-solid PCF and the proposed experimental setup for phase interrogation.

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We use finite element method to investigate the electromagnetic field distribution in the fiber sensor. Propagation constants are calculated in modal analysis. The electromagnetic field in the out-of-plane direction is defined as:

E(r,t)=Re(E˜(r)eζz+iωt)=Re(E˜(r)ek0Im(neff)zik0Re(neff)z+iωt
where ζ is the eigenvalue, ω and k0 are the frequency and wave vector of the input light, and neff is the effective refractive index. In the exponential factor of Eq. (1), the first term characterizes the attenuation, and the second and the third terms are the sinusoidal phase change along the propagation axis and with time.

3. Simulation results

3.1. Plasmonic wave excitation for different polishing depths

Since there has been little attention paid to phase modulation in D-shaped fiber SPR sensors, we would like to first present an overview of the associated phenomena under deep polishing conditions and discuss the underlying physics. Figure 2 shows the numerically calculated plasmonic wave stimulation at the sensor surface for different polishing depths: from top to bottom, 1.3Λ, 1.2Λ, 1.1Λ, 1.0Λ, and 0.9Λ. The images on the left show the spatial distribution of the electric field density at a wavelength near the phase match point. Plasmonic waves are stimulated by the core mode leakage and propagate along the metal surface. It is clear that the closer the metal surface is to the fiber center, the more intense is the plasmonic wave. The plots on the right side present the phase match condition of the plasmonic mode and the core mode under the condition ne = 1.333 (for water). Each plot includes four lines: the real part (blue) and imaginary part (green) of the effective refractive index (neff) for the resonant fundamental mode (HE11) at different wavelengths, the plasmonic mode (red dashed lines), as well as the real part of the effective refractive index (neff′) for the non-resonant fundamental modes (magenta). At the crossing point of the core mode line and the plasmonic mode line, the phase match condition is satisfied, and consequently a resonance occurs, causing the loss peak in the imaginary part of neff. The magenta lines will be discussed below, in the context of phase shift extraction.

 figure: Fig. 2

Fig. 2 Left: Electric field distribution and stimulation of plasmonic waves at wavelengths near λres, for different polishing plane positions h = 1.3Λ, 1.2Λ, 1.1Λ, 1.0Λ, 0.9Λ, respectively. Right: Phase match condition for each of the five cases. The imaginary part of neff (green curve) exhibits a peak at the crossing point of th core mode (blue) and the plasmonic mode (red dashed lines). The magenta lines are the real part of the fundamental mode for non-resonance cases.

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On the right side of Fig. 2, the blue curves for Re(neff) display S-shaped kinks. As the polishing depth increases, the kink becomes more pronounced. These kinks expose the underlying physics of surface plasmon resonance, where the phase of the transmitted electromagnetic field is modified by the resonance. The effective refractive index neff represents the refractive index of the overall environment ‘perceived’ by the electromagnetic wave. In general, as the wavelength increases, Re(neff) decreases since the electromagnetic field will penetrate deeper into the cladding region where the refractive index is lower. When a resonance takes place, free electrons are driven by the evanescent fields, and the oscillation of resonant electrons will react upon the fields, influencing their propagation. The modes of the fiber are modified and their phase velocity changes. The ‘slow’ electromagnetic field – with wavelength shorter than the resonance wavelength – is accelerated by the resonance, while the ‘fast’ portion – with wavelength longer than the resonance wavelength – is decelerated. Therefore, the shape of the effective refractive index around the resonance wavelength is distorted into the observed S-shape kink. The more intense the resonance, the more pronounced the curve. We have found an indication of this distortion in [22] Fig. 6(b), which is consistent with our results here.

To show the details of the resonances, in Fig. 3 we quantify the electric field density along the y-axis and their loss properties for different polishing depths. The inset in Fig. 3(b) zooms in on the electric field density around the leakage channel. We denote the maximum electric field density on the metallic surface as Emax. As the polishing plane is lowered, the surface electric field is significantly enhanced. Except for a slight lowering of the center of the mode, the general shapes of the curves overlap. This indicates that the leakage has little impact on the modal distribution. Figure 3(c) shows that the maximum surface electric field can be fitted by an exponential function of the metal film’s position. We will discuss the relevance of this result to the sensor performance later.

 figure: Fig. 3

Fig. 3 (a) The general shape of fiber SPR sensor based on all-solid PCF. Red line indicates the y-axis. (b) Electric field density along the y-axis. Inset focuses in the region of around leakage channel. (c) The fitted result of the maximum surface electric fields (Emax) with respect to the height of the polishing plane. (d) The loss of fiber SPR sensors with different polishing depths are presented. We prefer fiber sensor with h = 1.1Λ for the experimental investigation.

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In Fig. 3(d), the losses for fiber SPR sensors with different polishing depths are presented, which will help us to determine the sensor parameter. Referring to the report [24], we present an experimental setup for a phase-sensitive fiber sensor in the lower half of Fig. 1. The incident coherent light is split into a vertical component and a horizontal component with respect to the polishing plane. By injecting a periodic phase (ϕ(t)), the phase difference between the two electrical field components can be extracted. The measured intensity at the photodiode can be written as

Iout=EE*=Ip+Is+2IpIscos(ω0t+θ)IDC+2(12kpIp0)(12ksIs0)cos(ωt+θ)IDC+kpI0cos(ωt+θ)
where kp and ks are the attenuation coefficients of the p-wave and s-wave, ω0 is the frequency injected by the phase retarder and θ is the phase shift caused by the resonance. The factor of 1/2 accounts for the effect of the 45° polarization analyzer. A critical condition for the feasibility of the experimental setup is that the cosine term with phase information be detectable for wave reconstruction. Thus this term should be greater than the noise level of the photodiode. Considering a detection amplitude of −65 dBm for the cosine term, and original intensities for the two beams of 0 dBm, the affordable loss for the sensing beam is Lp = I0kpI0 = 130 dB, doubled for the square root symbol. For a typical 1–3 cm sensing length, the polishing height h = 1.1λ with 50 dB loss would be preferred. Even higher losses with a shorter sensing region could be tolerated as long as the phase signal conveyed by the p-wave can be demodulated from the interference.

3.2. Phase extraction scheme

As mentioned above, we are interested in the performance of a D-shaped fiber SPR based on photonic crystal fiber under phase interrogation. In the upper half of Fig. 1, the interface, the rods, the plasmonic wave and the core mode are depicted. In contrast to SM fiber [21], all-solid photonic fiber has a “blurred” core/cladding interface, which separates the core mode and the SPR wave. As illustrated, in a D-shaped photonic crystal fiber sensor, the fiber rods act like stumps; they act not only as a cladding to confine the core mode, but also provide a passage channel for a field leakage. The above numerical results have demonstrated the effect of the distance between the metallic film and the fiber core (i.e. the length of the leakage channel) on the magnitude of the evanescent field. The high-refractive-index leakage channel plays a key role in guiding the direction and tuning the amplitude of the field leakage.

For the experimental setup described in Fig. 1 and using Eq. (1), the phase shift is obtained simply from the difference between Re(neff) of two modes. The phase difference is thus defined as

Φd=2πλ(Re(np)Re(ns))L
where λ is the incident wavelength, L is the length of sensor region, Re(np) is the real part of the effective refractive index for SPR cases (the blue lines in Fig. 2), and Re(ns) represents that in the non-resonance cases (the magenta lines). The existence of L means that the phase sensitivity is a linear function of the sensing length, in contrast to spectral methods.

We present the phase extraction scheme in Fig. 4 for the specific case of a sensor with polishing depth h = 1.1Λ. Figure 4(a) shows the phase difference Φd and its shift for various environmental refractive indices (ne). The reduction at long wavelengths is due to the significant birefringence of the D-shaped fiber. Figure 4(b) focuses on the resonance region of Fig. 4(a). Vertical lines represent the wavelengths of the incident light, and their crossing points with the curves give the phase differences for each ne. Multiple lines correspond to a wavelength multiplex scheme, such as that implemented in the Kretschmann configuration [27]. By tuning the wavelength of the incident light, the measurement range can be extended. Figure 4(c) is a plot of the phase shift for incident light wavelengths λ1 = 600 nm, λ2 = 612 nm and λ3 = 620 nm. The dashed black boxes highlight the quasi-linear region. Note that the sign of Φd is reversed here relative to Fig. 4(b) for clarity. Finally, in Fig. 4(d) the solid curves present the sensitivities for different ne and the dashed curves present their losses. The heavy curves correspond to the quasi-linear regions highlighted in Fig. 4(c).

 figure: Fig. 4

Fig. 4 (a) The phase difference of two modes Φd under different conditions. (b) Enlarged drawing for the resonance region in (a). Vertical lines represent the wavelengths of the incident light for a wavelength multiplex scheme. Different colors represent different wavelengths. (c) The phase shift for incident light wavelengths as ne changes. (d) The solid curves present the sensitivities and the dashed curves show the corresponding loss properties. The heavy lines highlight the results from the dashed black boxes in (c).

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The sensitivity is found to depend strongly on the incident light wavelength. Thus a multiplex scheme, with a tunable incident light wavelength, is important in phase interrogation. Meanwhile, as the environmental refractive index ne increases, the resonance wavelength shifts toward a longer wavelength and the leakage of electromagnetic wave is intensified, which leads to an even higher sensitivity and sensor loss. The red curve present a sensitivity of 3.3 × 104 Deg/RIU/cm at ne = 1.340 with maximum fiber loss 60dB/cm. Considering a 130 dB loss in the sensing region, the sensing length is 2.17 cm long and its overall sensitivity is 7.15 × 104 Deg/RIU. The predicted sensitivity (Δϕne) in the blue curve is greater than 5.0 × 104 deg/RIU/cm around the refractive index 1.348, while its maximum loss is about 80 dB/cm. Similarly, the overall sensitivity is 8.12 × 104 deg/RIU. For the green curves at ne = 1.353 RIU, the sensitivity is about 6.5 × 104 deg/RIU/cm and the loss is 93dB/cm. So the overall sensitivity is approximately 9.09 × 104 deg/RIU. In summary, although constrained by the sensor loss, the final sensitivity is great than 7 × 104 Deg/RIU and is improved slowly as ne increases. A simple extrapolation suggests a sensitivity of 2.17 × 105 deg/RIU at ne = 1.44 RIU. This result is much higher than the experimental result 5.7 × 104 Deg/RIU in [24], in which authors may exhaust the loss property of the fiber sensor by tuning the environmental refractive index, but the sensitivity is bound by an improper wavelength. Therefore, due to the shift of loss property and sensitivity when the surface mass load increases, the design of the sensor, with choice of the parameters – such as the wavelength, polishing depth, and sensing length –tailored to the specific application, is important for delivering optimal performance.

Having demonstrated the principle of a phase interrogation fiber SPR sensor, we will now discuss variations to the fiber sensor design and their effect on its sensitivity. This time we will focus on the connection between the structural variance, the evanescent field and the sensor performance, without taking loss properties into consideration.

3.3. Influence of the polishing depth on sensor performance

As was demonstrated in Fig. 2, deeper polishing results in stronger resonance, as manifested by a more pronounced S-shape kink in the phase modulation curve and evident surface field enhancement. As a consequence, the phase shift is significantly influenced by the polishing depth. Figure 5(a) plots the phase difference curves for different polishing depths. Different colors represent different polishing depths, while different marker types represent different values for the environmental refractive indices ne = 1.333 − 1.345. Figure 5(b) shows the resonance wavelength λres for ne = 1.339, as a function of polishing depth. Figure 5(c) presents the phase shift curves with λres(1.339) for different ne. Best performance will be obtained in the quasi-linear region. Figure 5(d) plots the sensitivity based on linear fits to the three points closest to ne = 1.339. The latter sensitivity curve is fit very well by an exponential function that increases with the polishing depth. This compact result is consistent with the result in Fig. 3(c), demonstrating the strong correlation between the evanescent field and the sensor performance. Furthermore, it is clear that shallow polishing has a much smaller phase shift, which indicates that the guide zones at the ends of the sensing region should have only a minimal influence on the sensor performance.

 figure: Fig. 5

Fig. 5 (a) The phase difference curves for a range of different polishing depths. Different colors represent different polishing depths, while different marker types represent different values for the environmental refractive indices ne = 1.333 – 1.345. (b) The resonance wavelength λres(1.339) for different polishing depths. (c) The phase shift curves as ne increases. (d) The sensitivity for different polishing depths and the fitted result.

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3.4. Influence of the base fiber structure

The pitch and rod radius of the base fiber determine the field confinement; thus, these parameters should have a strong influence on the sensor performance. For convenience in the following discussion, we set the polishing depth at h = 1.1Λ and the film thickness to be t = 45 nm. In Fig. 6, the left side displays the effect of the pitch on the sensitivity, and the right side likewise for the cladding rod radius. Different colors represent different pitches/rod radii, and different markers represent different ne. The top row is the phase difference curves, and the second row shows the phase shift for different ne. The corresponding sensitivities are plotted in the third row. The fourth and bottom rows present the electric field along the y-axis and the fitted results, as for Fig. 3.

By comparing the results shown in Fig. 6, we find that the pitch of the base fiber is the most critical structural factor determining the sensor’s performance. As the pitch is decreased from 2.8 μm to 1.4 μm, the sensitivity improves nearly 70 times. The rod radius also has a significant influence on the sensor performance: the sensor performance increases from 17000 Deg/RIU/cm to 48100 Deg/RIU/cm when the rod radius is reduced from 0.48Λ to 0.32Λ.

 figure: Fig. 6

Fig. 6 Fiber structural parameters and their influence in sensor performance. The left side displays the impact of the pitch on the sensitivity, and the right side for the cladding rod radius. The structural variants are plotted in different color, while different marker types represent different ne. The top row is the phase difference curves, the second row shows the phase shift for different ne. The sensitivities are plotted in the third row. The fourth and bottom rows present the electric field along the y-axis and the corresponding fitted results. The red circle in (g) highlight the protrusion of the wave vector under small pitch condition Λ = 1.4μm.

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The electric field (normalized by its maximum value) is plotted along the y-axis for different values of the pitch in Fig. 6(g) and different rod radii in Fig. 6(h), revealing more details about the structural effects. As the pitch decreases, the surface electric field is significantly enhanced. For Λ = 1.4μm, the maximum electric field is even closer to the core mode. The general shapes of these curves are similar to each other, with the exception of the Λ = 1.4μm case (red line); in the region marked by the red circle at the foot of the peak in Fig. 6(g), the red line exhibits an obvious bump not present other lines, which suggests a protrusion of the wave vector. As the rod diameter decreases, the left side of the E profiles in Fig. 6(h) separate from each other, which suggests more electric field penetration into the gap space between the rods, while on the right side the field clings to the metallic film regardless on the rod diameter. Apparently, the structure of PCF with gaps enable the protrusion and the penetration. When r=0.32Λ, where the leakage channel is wide enough, the effect of field enhancement decreases, shown as an slight deviation from a linear extrapolation in (j).

So far, we have comprehensively investigated a model of a phase-sensitive D-shaped fiber SPR sensor based on an all-solid photonic crystal fiber. Our sensor design takes advantage of the structure of photonic crystal fiber. The field leakage is guided to penetrate through the gaps between the ‘stumps’ in the core and the film. Thus, the structural parameters of the base fiber play a fundamental role in the fiber sensor design, and the polishing depth is the key for finely tuning the plasmonic wave stimulation.

4. Discussion

A D-shaped all-solid fiber SPR sensor has many advantageous features, such as ease of fabrication, a planar surface for self-assembled mono-layer growth, and convenience for inspection. More essentially, the D-shaped cross section provides a directional leakage channel for the evanescent field. Meanwhile, the asymmetry of the two electromagnetic wave components helps to accomplish the phase extraction. Thus phase sensitivity and the channel-guide leakage are two key features of the D-shaped fiber SPR sensor based on PCF.

4.1. From wavelength interrogation to phase interrogation

An important discovery is that phase interrogation is always positively correlated with the resonance amplitude. This statement is not as trivial as it may appear. Figure 5(b) shows the shift of the resonance wavelength with polishing depth; when the polishing plane is lowered, a small or even negative shift occurs, whereas the resonance gets stronger. A similar situation occurs for the case of smaller pitch. Although the resonance wavelength shift is roughly proportional to the environment refractive index, it does not render the resonance amplitude. This suggests that the wavelength shift is only an overall effect of the internal phase modulation. In contrast, with a stronger resonance in the deeper polished configuration, fiber sensor working in phase interrogation mode attains a significant improvement. This difference make phase interrogation more sensitive than traditional wavelength interrogation. This conclusion is consistent with results from prism SPR studies.

To estimate the sensor performance, we focus on its sensitivity and the baseline noise. As was pointed out in Sec. 3.2, the sensitivity is strongly dependent on the input light and constrained by the sensor loss. Thus suppression of the baseline noise becomes important. As has been pointed out in [23], a suitable a detection scheme design is critical for attaining a better signal-to-noise ratio in phase interrogation. In the experimental configuration proposed in [24] and illustrated in Fig. 1, there are two measures to reduce the noise: modulation-demodulation processing for isolating the noise from the light source, and a counteraction effect from the phase difference where other environmental factors will act equivalently on both vertical and horizontal arms. These measures will help to decrease the noise level and provide a further order of magnitude improvement to the detection limit. However, the reported baseline noise in [24] was 0.4 Deg, which is much higher than that in prism reports [25]. The most obvious difference between the fiber sensor and prism experiments is the divergence of the light. The prism experiment used a laser source with 1 mrad divergence, while the critical angle of multi-mode fiber is about 23°. Apparently, the divergence of light is the primary factor for the baseline noise.

As emphasized in the introduction, we use PCF to replace the multi-mode fiber, which will limit the divergence of the propagating light. The numerical aperture of PCF is less than 0.22, corresponding to a critical angle of 12°, which is half of that in a multi-mode fiber. Thus we expect a noise level of 0.2 Deg for the D-shaped PCF sensor. The sensitivity is 9.09 × 104 Deg/RIU at ne = 1.353 with λ = 620 nm. Correspondingly, the resolution would be 2.2 × 10−6 RIU. This resolution is much closer to that of a Kretschmann configuration with wavelength interrogation. With this marked improvement in sensor performance, the applications of fiber SPR sensors will be greatly extended, enabling a wide range of biochemical interaction measurements to be carried out.

4.2. Significance of channel-guide leakage in D-shaped PCF fiber sensors

Since optical fiber works on total internal reflection and SPR sensing depends on the evanescent field, the critical issue in fiber SPR sensor design is how to simultaneously confine the core mode in the center of the fiber, while also providing efficient and tunable field leakage for optical sensing. Bearing this in mind, we proceed with the following discussion of the advantages afforded by our fiber design with regard to field leakage.

The geometry of our modeled D-shaped PCF fiber sensor is illustrated in Fig. 1. As a periodic structure, confinement in PCF is enhanced according to Bloch’s theorem. Leakage along the channel are shown in Figs. 2, 3 and 6. As shown, the leakage channel is the most effective pathway for outflow of the evanescent field, the magnitude of which is tunable via the postprocessing of the fiber.

To clarify the advantages of the D-shaped fiber sensor based PCF, we make a comparison with that based on a step fiber. Since it is hard to set a fair basis for comparison, we prefer a systematic investigation of fiber sensors with different polishing depths, which will elucidate the differences in leakage regulation and sensor performance inherent to the two fiber types. The cross section of the D-shaped fiber SPR sensor based on a step fiber is shown in Fig. 7(a). The diameter of the fiber core is 2.4 μm, which corresponds to the inner edge of the PCF for a strong evanescent field. Similar to Fig. 3, the evanescent field distributions along the y-axis for step fiber and PCF sensors are presented in Fig. 7(b). Two groups curves are separated: the solid line represents fiber sensors based on PCF, while the dashed line represents fiber sensor based on step fiber. Different colors corresponds to different depths. Clearly, deeper polishing position causes a stronger evanescent field. The maximum surface field is compared in Fig. 7(c), plotted on logarithmic axes to exhibit the differences between the two fiber types.

 figure: Fig. 7

Fig. 7 (a) The cross section of D-shaped step fiber sensor. The environmental refractive index is ne = 1.339 for comparison. (b) The evanescent field along the y-axis. The solid line represents fiber sensors based on PCF, while the dashed line represents fiber sensor based on step fiber. Different color corresponds to different depths. (c) Comparison of the maximum surface field (logarithmic axes). (d) The corresponding phase difference curves and sensitivities for ne = 1.339 RIU and polishing depth h = 1.1 Λ.

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Apparently, the leakage of the PCF sensor is higher under same condition and its slope is gentler. This is natural since the leakage channel is of high refractive index, the same as that of the fiber core. Thus the leakage is much easier than that in the step fiber sensor. This discrepancy is highly significant when the polishing is shallow. Meanwhile, not being modified by the interface, the slope of the exponential function is smaller, corresponding to a slower change velocity. This is useful in sensor fabrication since the smaller slope means that the same loss range will correspond to a larger range of variance in polishing position. With this analysis of the evanescent field, their difference in performance in Fig. 7(d) is reasonable, where the sensitivity of PCF sensor is nearly twice that of the step fiber sensor. Obviously, the channel guided leakage is efficient and convenient in tunning, which is consistent to the guideline in SPR sensor design. Therefore, we have demonstrated that the D-shaped fiber SPR sensor based on PCF is a better choice than that based on step fibers.

We believe these differences arise from the different leakage regulation mechanisms of the two fiber types, where the structural parameters play important roles. So we analyze below the structural effects of PCF on the evanescent field, and model its impact on the sensor performance.

4.3. Structural effects on the evanescent field

In Fig. 6, we have fitted both the sensitivity and Emax with respect to the structure parameters of the fiber. These results allow us to make some connections between the evanescent field and the sensor performance. The sensitivity and Emax are both linear functions of the rod diameter (Fig. 6(f) and (j) respectively), which suggests a linear relation between the evanescent field and the sensor performance. However, for different pitches and polishing plane positions, the sensitivity and Emax vary exponentially, according to

S=a1exp(b1x)Emax=a2exp(b2x)
where S stands for the sensitivity. It follows that
S=a2a1(b1/b2)Emax(b1/b2)
This result suggests that the sensitivity is a power law of Emax with an exponent equal to the ratio of coefficients that are extracted from the exponential fits of S and Emax, as shown in Fig. 6(e) and (i) with pitch as the variable. This exponent turns out to be k1 = (−3.3489)/(−1.1006) = 3.0428 for pitch variation and k2 = (−6.8723)/(−3.2044)= 2.1446 for polishing plane depth. These three structural parameters – the pitch of PCF, the height of the polishing plane and the rod diameter – thus vary significantly in their impact on the evanescent field. Apparently, to tune the sensor performance and the corresponding loss, changing the pitch is the most effective strategy, and varying the rod diameter the least effective.

Furthermore, the variation of the rod diameter changes the width of the leakage channel, consequently enhancing the evanescent field, and resulting in a proportional improvement in the sensitivity. When the polishing plane is closer to the core, obviously the stimulated field on the film should be enforced, exponentially enhancing the evanescent field and the sensor performance. As for changes to the pitch, shrinking the fiber core will induce strong squeezing on the core mode, which would be a much more effective way of boosting the evanescent field than the other methods. Certainly, there are differences in the exponent for the sensitivity dependence on evanescent field and on Emax; this may be explained as a complex relationship between the overall effect of the evanescent field and the maximum point since the surface electromagnetic field is propagating along a striped surface.

These results inspire us to model the relationship between the sensor configuration, Emax, the evanescent field, and the sensor performance. Since the function of the sensor depends on the evanescent field, and the sensitivity is such a simple function of the structural variables, we assume there is an intermediate variable Γ between the structural variables and the sensitivity, and the sensitivity is a simple linear function of Γ. So Γ represents the overall effect on the evanescent field, whereas it is correlated to the point value Emax.

Γ=EdSΓ(Emax)
In brief, we get the following relation
structuralvariableΓ(Emax)sensitivity
Therefore, we generalize the effect of the sensor structural variables into Γ(Emax). The overall effect caused by structural variables are different. Specifically, in Eq. (6) Γ(Emax) is a linear function of Emax for changes in the rod diameter, quasi-two dimensional for the polishing position, and close to three dimensional in the case of the pitch variable.

5. Conclusion

In summary, we numerically investigated a deep-polished fiber SPR sensor design based on all-solid photonic crystal fiber, and found that it exhibits significant phase modulation. By exploiting the phase modulation and adopting phase interrogation, the sensor system can attain sensitivity levels markedly superior to traditional wavelength interrogation techniques. We assessed the model sensor performance for a range of different structural parameters. A comparison with a D-shaped step fiber sensor design suggests that the leakage of PCF sensors is more efficient and tunable, a consequence of its flexible structure. A mathematical model is proposed to describe the effects of structural variables on the evanescent field and the sensor performance.

Acknowledgments

The authors would like to thank Dr. Scott Edwards, Shenzhen University, for his kind help in language editing. This work was supported in part by the National Science Foundation of China under Grant (No. 61275125, No. 11375117, and No. 61308046), Basic Research Program of Shenzhen and High-level Talents Project of Guangdong Province.

References and links

1. J. Homola, Surface Plasmon Resonance Based Sensors, (Springer, 2006). [CrossRef]  

2. R. Jorgenson and S. Yee, “A fiber-optic chemical sensor based on surface plasmon resonance,” Sens. Actuat. B-Chem. 12, 213–220 (1993). [CrossRef]  

3. S. A. Kim, S. J. Kim, H. Moon, and S. B. Jun, “In vivo optical neural recording using fiber-based surface plasmon resonance,” Opt. Lett. 37, 614–616 (2012). [CrossRef]   [PubMed]  

4. F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012). [CrossRef]   [PubMed]  

5. Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010). [CrossRef]  

6. H. W. Lee, M. A. Schmidt, and P. S. Russell, “Excitation of a nanowire molecule in gold-filled photonic crystal fiber,” Opt. Lett. 37, 2946–2948 (2012). [CrossRef]   [PubMed]  

7. P. Uebel, M. A. Schmidt, H. W. Lee, and P. S. Russell, “Polarisation-resolved near-field mapping of a coupled gold nanowire array,” Opt. Express 20, 28409–28417 (2012). [CrossRef]   [PubMed]  

8. A. Hassani and M. Skorobogatiy, “Design of the microstructured optical fiber-based surface plasmon resonance sensors with enhanced microfluidics,” Opt. Express 14, 11616–11621 (2006). [CrossRef]   [PubMed]  

9. M. Hautakorpi, M. Mattinen, and H. Ludvigsen, “Surface-plasmon-resonance sensor based on three-hole microstructured optical fiber,” Opt. Express 16, 8427–8432 (2008). [CrossRef]   [PubMed]  

10. Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012). [CrossRef]   [PubMed]  

11. P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012). [CrossRef]   [PubMed]  

12. J. Li, L.-P. Sun, S. Gao, Z. Quan, Y.-L. Chang, Y. Ran, L. Jin, and B.-O. Guan, “Ultrasensitive refractive-index sensors based on rectangular silica microfibers,” Opt. Lett. 36, 3593–3595 (2011). [CrossRef]   [PubMed]  

13. S. Lepinay, A. Staff, A. Ianoul, and J. Albert, “Improved detection limits of protein optical fiber biosensors coated with gold nanoparticles,” Biosens. Bioelectron. 52, 337–344 (2014). [CrossRef]  

14. M. Alam and J. Albert, “Selective excitation of radially and azimuthally polarized optical fiber cladding modes,” J. Lightwave Technol. 31, 3167–3175 (2013). [CrossRef]  

15. J. Albert, L.-Y. Shao, and C. Caucheteur, “Tilted fiber bragg grating sensors,” Laser Photon. Rev. 7, 83–108 (2013). [CrossRef]  

16. Y. Shevchenko, C. Chen, M. A. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Lett. 35, 637–639 (2010). [CrossRef]   [PubMed]  

17. M.-H. Chiu, S.-F. Wang, and R.-S. Chang, “D-type fiber biosensor based on surface-plasmon resonance technology and heterodyne interferometry,” Opt. Lett. 30, 233–235 (2005). [CrossRef]   [PubMed]  

18. Y. Zhang, C. Gu, A. M. Schwartzberg, and J. Z. Zhang, “Surface-enhanced raman scattering sensor based on d-shaped fiber,” Appl. Phys. Lett. 87, 123105 (2005). [CrossRef]  

19. H.-Y. Lin, W.-H. Tsai, Y.-C. Tsao, and B.-C. Sheu, “Side-polished multimode fiber biosensor based on surface plasmon resonance with halogen light,” Appl. Opt. 46, 800–806 (2007). [CrossRef]   [PubMed]  

20. Z. Tan, X. Li, Y. Chen, and P. Fan, “Improving the sensitivity of fiber surface plasmon resonance sensor by filling liquid in a hollow core photonic crystal fiber,” Plasmonics 9, 167–173 (2014). [CrossRef]  

21. Y. J. He, “Novel d-shape lspr fiber sensor based on nano-metal strips,” Opt. Express 21, 23498–23510 (2013). [CrossRef]   [PubMed]  

22. M. Tian, P. Lu, L. Chen, C. Lv, and D. Liu, “All-solid d-shaped photonic fiber sensor based on surface plasmon resonance,” Opt. Commun. 285, 1550–1554 (2012). [CrossRef]  

23. A. V. Kabashin, S. Patskovsky, and A. N. Grigorenko, “Phase and amplitude sensitivities in surface plasmon resonance bio and chemical sensing,” Opt. Express 17, 21191–21204 (2009). [CrossRef]   [PubMed]  

24. Y.-L. Lo, C.-H. Chuang, and Z.-W. Lin, “Ultrahigh sensitivity polarimetric strain sensor based upon d-shaped optical fiber and surface plasmon resonance technology,” Opt. Lett. 36, 2489–2491 (2011). [CrossRef]   [PubMed]  

25. S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29, 2378–2380 (2004). [CrossRef]   [PubMed]  

26. A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005). [CrossRef]  

27. Y. Shao, Y. Li, D. Gu, K. Zhang, J. Qu, J. He, X. Li, S.-Y. Wu, H.-P. Ho, M. G. Somekh, and H. Niu, “Wavelength-multiplexing phase-sensitive surface plasmon imaging sensor,” Opt. Lett. 38, 1370–1372 (2013). [CrossRef]   [PubMed]  

References

  • View by:

  1. J. Homola, Surface Plasmon Resonance Based Sensors, (Springer, 2006).
    [Crossref]
  2. R. Jorgenson and S. Yee, “A fiber-optic chemical sensor based on surface plasmon resonance,” Sens. Actuat. B-Chem. 12, 213–220 (1993).
    [Crossref]
  3. S. A. Kim, S. J. Kim, H. Moon, and S. B. Jun, “In vivo optical neural recording using fiber-based surface plasmon resonance,” Opt. Lett. 37, 614–616 (2012).
    [Crossref] [PubMed]
  4. F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012).
    [Crossref] [PubMed]
  5. Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
    [Crossref]
  6. H. W. Lee, M. A. Schmidt, and P. S. Russell, “Excitation of a nanowire molecule in gold-filled photonic crystal fiber,” Opt. Lett. 37, 2946–2948 (2012).
    [Crossref] [PubMed]
  7. P. Uebel, M. A. Schmidt, H. W. Lee, and P. S. Russell, “Polarisation-resolved near-field mapping of a coupled gold nanowire array,” Opt. Express 20, 28409–28417 (2012).
    [Crossref] [PubMed]
  8. A. Hassani and M. Skorobogatiy, “Design of the microstructured optical fiber-based surface plasmon resonance sensors with enhanced microfluidics,” Opt. Express 14, 11616–11621 (2006).
    [Crossref] [PubMed]
  9. M. Hautakorpi, M. Mattinen, and H. Ludvigsen, “Surface-plasmon-resonance sensor based on three-hole microstructured optical fiber,” Opt. Express 16, 8427–8432 (2008).
    [Crossref] [PubMed]
  10. Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012).
    [Crossref] [PubMed]
  11. P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
    [Crossref] [PubMed]
  12. J. Li, L.-P. Sun, S. Gao, Z. Quan, Y.-L. Chang, Y. Ran, L. Jin, and B.-O. Guan, “Ultrasensitive refractive-index sensors based on rectangular silica microfibers,” Opt. Lett. 36, 3593–3595 (2011).
    [Crossref] [PubMed]
  13. S. Lepinay, A. Staff, A. Ianoul, and J. Albert, “Improved detection limits of protein optical fiber biosensors coated with gold nanoparticles,” Biosens. Bioelectron. 52, 337–344 (2014).
    [Crossref]
  14. M. Alam and J. Albert, “Selective excitation of radially and azimuthally polarized optical fiber cladding modes,” J. Lightwave Technol. 31, 3167–3175 (2013).
    [Crossref]
  15. J. Albert, L.-Y. Shao, and C. Caucheteur, “Tilted fiber bragg grating sensors,” Laser Photon. Rev. 7, 83–108 (2013).
    [Crossref]
  16. Y. Shevchenko, C. Chen, M. A. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Lett. 35, 637–639 (2010).
    [Crossref] [PubMed]
  17. M.-H. Chiu, S.-F. Wang, and R.-S. Chang, “D-type fiber biosensor based on surface-plasmon resonance technology and heterodyne interferometry,” Opt. Lett. 30, 233–235 (2005).
    [Crossref] [PubMed]
  18. Y. Zhang, C. Gu, A. M. Schwartzberg, and J. Z. Zhang, “Surface-enhanced raman scattering sensor based on d-shaped fiber,” Appl. Phys. Lett. 87, 123105 (2005).
    [Crossref]
  19. H.-Y. Lin, W.-H. Tsai, Y.-C. Tsao, and B.-C. Sheu, “Side-polished multimode fiber biosensor based on surface plasmon resonance with halogen light,” Appl. Opt. 46, 800–806 (2007).
    [Crossref] [PubMed]
  20. Z. Tan, X. Li, Y. Chen, and P. Fan, “Improving the sensitivity of fiber surface plasmon resonance sensor by filling liquid in a hollow core photonic crystal fiber,” Plasmonics 9, 167–173 (2014).
    [Crossref]
  21. Y. J. He, “Novel d-shape lspr fiber sensor based on nano-metal strips,” Opt. Express 21, 23498–23510 (2013).
    [Crossref] [PubMed]
  22. M. Tian, P. Lu, L. Chen, C. Lv, and D. Liu, “All-solid d-shaped photonic fiber sensor based on surface plasmon resonance,” Opt. Commun. 285, 1550–1554 (2012).
    [Crossref]
  23. A. V. Kabashin, S. Patskovsky, and A. N. Grigorenko, “Phase and amplitude sensitivities in surface plasmon resonance bio and chemical sensing,” Opt. Express 17, 21191–21204 (2009).
    [Crossref] [PubMed]
  24. Y.-L. Lo, C.-H. Chuang, and Z.-W. Lin, “Ultrahigh sensitivity polarimetric strain sensor based upon d-shaped optical fiber and surface plasmon resonance technology,” Opt. Lett. 36, 2489–2491 (2011).
    [Crossref] [PubMed]
  25. S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29, 2378–2380 (2004).
    [Crossref] [PubMed]
  26. A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
    [Crossref]
  27. Y. Shao, Y. Li, D. Gu, K. Zhang, J. Qu, J. He, X. Li, S.-Y. Wu, H.-P. Ho, M. G. Somekh, and H. Niu, “Wavelength-multiplexing phase-sensitive surface plasmon imaging sensor,” Opt. Lett. 38, 1370–1372 (2013).
    [Crossref] [PubMed]

2014 (2)

S. Lepinay, A. Staff, A. Ianoul, and J. Albert, “Improved detection limits of protein optical fiber biosensors coated with gold nanoparticles,” Biosens. Bioelectron. 52, 337–344 (2014).
[Crossref]

Z. Tan, X. Li, Y. Chen, and P. Fan, “Improving the sensitivity of fiber surface plasmon resonance sensor by filling liquid in a hollow core photonic crystal fiber,” Plasmonics 9, 167–173 (2014).
[Crossref]

2013 (4)

2012 (7)

M. Tian, P. Lu, L. Chen, C. Lv, and D. Liu, “All-solid d-shaped photonic fiber sensor based on surface plasmon resonance,” Opt. Commun. 285, 1550–1554 (2012).
[Crossref]

S. A. Kim, S. J. Kim, H. Moon, and S. B. Jun, “In vivo optical neural recording using fiber-based surface plasmon resonance,” Opt. Lett. 37, 614–616 (2012).
[Crossref] [PubMed]

F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012).
[Crossref] [PubMed]

H. W. Lee, M. A. Schmidt, and P. S. Russell, “Excitation of a nanowire molecule in gold-filled photonic crystal fiber,” Opt. Lett. 37, 2946–2948 (2012).
[Crossref] [PubMed]

P. Uebel, M. A. Schmidt, H. W. Lee, and P. S. Russell, “Polarisation-resolved near-field mapping of a coupled gold nanowire array,” Opt. Express 20, 28409–28417 (2012).
[Crossref] [PubMed]

Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012).
[Crossref] [PubMed]

P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (2)

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

Y. Shevchenko, C. Chen, M. A. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Lett. 35, 637–639 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

2007 (1)

2006 (1)

2005 (3)

M.-H. Chiu, S.-F. Wang, and R.-S. Chang, “D-type fiber biosensor based on surface-plasmon resonance technology and heterodyne interferometry,” Opt. Lett. 30, 233–235 (2005).
[Crossref] [PubMed]

Y. Zhang, C. Gu, A. M. Schwartzberg, and J. Z. Zhang, “Surface-enhanced raman scattering sensor based on d-shaped fiber,” Appl. Phys. Lett. 87, 123105 (2005).
[Crossref]

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

2004 (1)

1993 (1)

R. Jorgenson and S. Yee, “A fiber-optic chemical sensor based on surface plasmon resonance,” Sens. Actuat. B-Chem. 12, 213–220 (1993).
[Crossref]

Alam, M.

Albert, J.

S. Lepinay, A. Staff, A. Ianoul, and J. Albert, “Improved detection limits of protein optical fiber biosensors coated with gold nanoparticles,” Biosens. Bioelectron. 52, 337–344 (2014).
[Crossref]

M. Alam and J. Albert, “Selective excitation of radially and azimuthally polarized optical fiber cladding modes,” J. Lightwave Technol. 31, 3167–3175 (2013).
[Crossref]

J. Albert, L.-Y. Shao, and C. Caucheteur, “Tilted fiber bragg grating sensors,” Laser Photon. Rev. 7, 83–108 (2013).
[Crossref]

Y. Shevchenko, C. Chen, M. A. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Lett. 35, 637–639 (2010).
[Crossref] [PubMed]

Araki, A.

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

Barchiesi, D.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Caucheteur, C.

J. Albert, L.-Y. Shao, and C. Caucheteur, “Tilted fiber bragg grating sensors,” Laser Photon. Rev. 7, 83–108 (2013).
[Crossref]

Chang, R.-S.

Chang, Y.-L.

Chapelle, M.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Chen, C.

Chen, L.

M. Tian, P. Lu, L. Chen, C. Lv, and D. Liu, “All-solid d-shaped photonic fiber sensor based on surface plasmon resonance,” Opt. Commun. 285, 1550–1554 (2012).
[Crossref]

Chen, Y.

Z. Tan, X. Li, Y. Chen, and P. Fan, “Improving the sensitivity of fiber surface plasmon resonance sensor by filling liquid in a hollow core photonic crystal fiber,” Plasmonics 9, 167–173 (2014).
[Crossref]

Chiu, M.-H.

Chuang, C.-H.

Dakka, M. A.

Delport, F.

F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012).
[Crossref] [PubMed]

Fan, P.

Z. Tan, X. Li, Y. Chen, and P. Fan, “Improving the sensitivity of fiber surface plasmon resonance sensor by filling liquid in a hollow core photonic crystal fiber,” Plasmonics 9, 167–173 (2014).
[Crossref]

Fu, X.-Y.

Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012).
[Crossref] [PubMed]

Gao, S.

Grigorenko, A. N.

Grimault, A.-S.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Gu, C.

Y. Zhang, C. Gu, A. M. Schwartzberg, and J. Z. Zhang, “Surface-enhanced raman scattering sensor based on d-shaped fiber,” Appl. Phys. Lett. 87, 123105 (2005).
[Crossref]

Gu, D.

Guan, B.-O.

Hao, C.-J.

Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012).
[Crossref] [PubMed]

Hassani, A.

Hautakorpi, M.

He, J.

He, Y. J.

Hide, M.

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

Hiragun, T.

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

Ho, H. P.

Ho, H.-P.

Homola, J.

J. Homola, Surface Plasmon Resonance Based Sensors, (Springer, 2006).
[Crossref]

Huang, X.-H.

Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012).
[Crossref] [PubMed]

Ianoul, A.

S. Lepinay, A. Staff, A. Ianoul, and J. Albert, “Improved detection limits of protein optical fiber biosensors coated with gold nanoparticles,” Biosens. Bioelectron. 52, 337–344 (2014).
[Crossref]

Janssen, K.

F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012).
[Crossref] [PubMed]

Jin, L.

Jorgenson, R.

R. Jorgenson and S. Yee, “A fiber-optic chemical sensor based on surface plasmon resonance,” Sens. Actuat. B-Chem. 12, 213–220 (1993).
[Crossref]

Jun, S. B.

Kabashin, A. V.

Kim, S. A.

Kim, S. J.

Kimura, T.

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

Knez, K.

F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012).
[Crossref] [PubMed]

Kong, S. K.

Lammertyn, J.

F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012).
[Crossref] [PubMed]

Law, W. C.

Lee, H. W.

Lepinay, S.

S. Lepinay, A. Staff, A. Ianoul, and J. Albert, “Improved detection limits of protein optical fiber biosensors coated with gold nanoparticles,” Biosens. Bioelectron. 52, 337–344 (2014).
[Crossref]

Li, J.

Li, X.

Z. Tan, X. Li, Y. Chen, and P. Fan, “Improving the sensitivity of fiber surface plasmon resonance sensor by filling liquid in a hollow core photonic crystal fiber,” Plasmonics 9, 167–173 (2014).
[Crossref]

Y. Shao, Y. Li, D. Gu, K. Zhang, J. Qu, J. He, X. Li, S.-Y. Wu, H.-P. Ho, M. G. Somekh, and H. Niu, “Wavelength-multiplexing phase-sensitive surface plasmon imaging sensor,” Opt. Lett. 38, 1370–1372 (2013).
[Crossref] [PubMed]

Li, Y.

Lin, C.

Lin, H.-Y.

Lin, Z.-W.

Liu, D.

M. Tian, P. Lu, L. Chen, C. Lv, and D. Liu, “All-solid d-shaped photonic fiber sensor based on surface plasmon resonance,” Opt. Commun. 285, 1550–1554 (2012).
[Crossref]

Lo, Y.-L.

Lu, P.

M. Tian, P. Lu, L. Chen, C. Lv, and D. Liu, “All-solid d-shaped photonic fiber sensor based on surface plasmon resonance,” Opt. Commun. 285, 1550–1554 (2012).
[Crossref]

Lu, Y.

Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012).
[Crossref] [PubMed]

Ludvigsen, H.

Lv, C.

M. Tian, P. Lu, L. Chen, C. Lv, and D. Liu, “All-solid d-shaped photonic fiber sensor based on surface plasmon resonance,” Opt. Commun. 285, 1550–1554 (2012).
[Crossref]

Macias, D.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Mattinen, M.

Moon, H.

Nakatani, T.

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

Niu, H.

Okamoto, K.

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

Patskovsky, S.

Pollet, J.

F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012).
[Crossref] [PubMed]

Qu, J.

Quan, Z.

Ran, Y.

Russell, P. S.

Schmidt, M. A.

Schwartzberg, A. M.

Y. Zhang, C. Gu, A. M. Schwartzberg, and J. Z. Zhang, “Surface-enhanced raman scattering sensor based on d-shaped fiber,” Appl. Phys. Lett. 87, 123105 (2005).
[Crossref]

Shao, L.-Y.

J. Albert, L.-Y. Shao, and C. Caucheteur, “Tilted fiber bragg grating sensors,” Laser Photon. Rev. 7, 83–108 (2013).
[Crossref]

Shao, Y.

Sheu, B.-C.

Shevchenko, Y.

Skorobogatiy, M.

Somekh, M. G.

Spasic, D.

F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012).
[Crossref] [PubMed]

Staff, A.

S. Lepinay, A. Staff, A. Ianoul, and J. Albert, “Improved detection limits of protein optical fiber biosensors coated with gold nanoparticles,” Biosens. Bioelectron. 52, 337–344 (2014).
[Crossref]

Sun, L.-P.

Suzuki, H.

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

Tan, Z.

Z. Tan, X. Li, Y. Chen, and P. Fan, “Improving the sensitivity of fiber surface plasmon resonance sensor by filling liquid in a hollow core photonic crystal fiber,” Plasmonics 9, 167–173 (2014).
[Crossref]

Tian, M.

M. Tian, P. Lu, L. Chen, C. Lv, and D. Liu, “All-solid d-shaped photonic fiber sensor based on surface plasmon resonance,” Opt. Commun. 285, 1550–1554 (2012).
[Crossref]

Tong, L.

P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
[Crossref] [PubMed]

Tsai, W.-H.

Tsao, Y.-C.

Tsutsui, T.

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

Uebel, P.

Verbruggen, B.

F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012).
[Crossref] [PubMed]

Vial, A.

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

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P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
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Wang, S.-F.

Wen, W.-Q.

Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012).
[Crossref] [PubMed]

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Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012).
[Crossref] [PubMed]

Wu, S. Y.

Wu, S.-Y.

Xia, Y.

P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
[Crossref] [PubMed]

Xu, X.

P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
[Crossref] [PubMed]

Yanase, Y.

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

Yao, J.-Q.

Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012).
[Crossref] [PubMed]

Yee, S.

R. Jorgenson and S. Yee, “A fiber-optic chemical sensor based on surface plasmon resonance,” Sens. Actuat. B-Chem. 12, 213–220 (1993).
[Crossref]

Ying, Y.

P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
[Crossref] [PubMed]

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Y. Zhang, C. Gu, A. M. Schwartzberg, and J. Z. Zhang, “Surface-enhanced raman scattering sensor based on d-shaped fiber,” Appl. Phys. Lett. 87, 123105 (2005).
[Crossref]

Zhang, K.

Zhang, L.

P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
[Crossref] [PubMed]

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Y. Zhang, C. Gu, A. M. Schwartzberg, and J. Z. Zhang, “Surface-enhanced raman scattering sensor based on d-shaped fiber,” Appl. Phys. Lett. 87, 123105 (2005).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Y. Zhang, C. Gu, A. M. Schwartzberg, and J. Z. Zhang, “Surface-enhanced raman scattering sensor based on d-shaped fiber,” Appl. Phys. Lett. 87, 123105 (2005).
[Crossref]

Biosens. Bioelectron. (2)

Y. Yanase, A. Araki, H. Suzuki, T. Tsutsui, T. Kimura, K. Okamoto, T. Nakatani, T. Hiragun, and M. Hide, “Development of an optical fiber {SPR} sensor for living cell activation,” Biosens. Bioelectron. 25, 1244–1247 (2010).
[Crossref]

S. Lepinay, A. Staff, A. Ianoul, and J. Albert, “Improved detection limits of protein optical fiber biosensors coated with gold nanoparticles,” Biosens. Bioelectron. 52, 337–344 (2014).
[Crossref]

J. Lightwave Technol. (1)

Laser Photon. Rev. (1)

J. Albert, L.-Y. Shao, and C. Caucheteur, “Tilted fiber bragg grating sensors,” Laser Photon. Rev. 7, 83–108 (2013).
[Crossref]

Nano Lett. (1)

P. Wang, L. Zhang, Y. Xia, L. Tong, X. Xu, and Y. Ying, “Polymer nanofibers embedded with aligned gold nanorods: a new platform for plasmonic studies and optical sensing,” Nano Lett. 12, 3145–3150 (2012).
[Crossref] [PubMed]

Nanotechnology (1)

F. Delport, J. Pollet, K. Janssen, B. Verbruggen, K. Knez, D. Spasic, and J. Lammertyn, “Real-time monitoring of dna hybridization and melting processes using a fiber optic sensor,” Nanotechnology 23, 065503 (2012).
[Crossref] [PubMed]

Opt. Commun. (1)

M. Tian, P. Lu, L. Chen, C. Lv, and D. Liu, “All-solid d-shaped photonic fiber sensor based on surface plasmon resonance,” Opt. Commun. 285, 1550–1554 (2012).
[Crossref]

Opt. Express (5)

Opt. Lett. (8)

S. A. Kim, S. J. Kim, H. Moon, and S. B. Jun, “In vivo optical neural recording using fiber-based surface plasmon resonance,” Opt. Lett. 37, 614–616 (2012).
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H. W. Lee, M. A. Schmidt, and P. S. Russell, “Excitation of a nanowire molecule in gold-filled photonic crystal fiber,” Opt. Lett. 37, 2946–2948 (2012).
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Y. Shevchenko, C. Chen, M. A. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Lett. 35, 637–639 (2010).
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M.-H. Chiu, S.-F. Wang, and R.-S. Chang, “D-type fiber biosensor based on surface-plasmon resonance technology and heterodyne interferometry,” Opt. Lett. 30, 233–235 (2005).
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Y.-L. Lo, C.-H. Chuang, and Z.-W. Lin, “Ultrahigh sensitivity polarimetric strain sensor based upon d-shaped optical fiber and surface plasmon resonance technology,” Opt. Lett. 36, 2489–2491 (2011).
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S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, and S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29, 2378–2380 (2004).
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Y. Shao, Y. Li, D. Gu, K. Zhang, J. Qu, J. He, X. Li, S.-Y. Wu, H.-P. Ho, M. G. Somekh, and H. Niu, “Wavelength-multiplexing phase-sensitive surface plasmon imaging sensor,” Opt. Lett. 38, 1370–1372 (2013).
[Crossref] [PubMed]

Phys. Rev. B (1)

A. Vial, A.-S. Grimault, D. Macias, D. Barchiesi, and M. Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Plasmonics (1)

Z. Tan, X. Li, Y. Chen, and P. Fan, “Improving the sensitivity of fiber surface plasmon resonance sensor by filling liquid in a hollow core photonic crystal fiber,” Plasmonics 9, 167–173 (2014).
[Crossref]

Sens. Actuat. B-Chem. (1)

R. Jorgenson and S. Yee, “A fiber-optic chemical sensor based on surface plasmon resonance,” Sens. Actuat. B-Chem. 12, 213–220 (1993).
[Crossref]

Sensors (1)

Y. Lu, C.-J. Hao, B.-Q. Wu, X.-H. Huang, W.-Q. Wen, X.-Y. Fu, and J.-Q. Yao, “Grapefruit fiber filled with silver nanowires surface plasmon resonance sensor in aqueous environments,” Sensors 12, 12016–12025 (2012).
[Crossref] [PubMed]

Other (1)

J. Homola, Surface Plasmon Resonance Based Sensors, (Springer, 2006).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Enlarged schematic of fiber SPR sensor based on an all-solid PCF and the proposed experimental setup for phase interrogation.
Fig. 2
Fig. 2 Left: Electric field distribution and stimulation of plasmonic waves at wavelengths near λres, for different polishing plane positions h = 1.3Λ, 1.2Λ, 1.1Λ, 1.0Λ, 0.9Λ, respectively. Right: Phase match condition for each of the five cases. The imaginary part of neff (green curve) exhibits a peak at the crossing point of th core mode (blue) and the plasmonic mode (red dashed lines). The magenta lines are the real part of the fundamental mode for non-resonance cases.
Fig. 3
Fig. 3 (a) The general shape of fiber SPR sensor based on all-solid PCF. Red line indicates the y-axis. (b) Electric field density along the y-axis. Inset focuses in the region of around leakage channel. (c) The fitted result of the maximum surface electric fields (Emax) with respect to the height of the polishing plane. (d) The loss of fiber SPR sensors with different polishing depths are presented. We prefer fiber sensor with h = 1.1Λ for the experimental investigation.
Fig. 4
Fig. 4 (a) The phase difference of two modes Φd under different conditions. (b) Enlarged drawing for the resonance region in (a). Vertical lines represent the wavelengths of the incident light for a wavelength multiplex scheme. Different colors represent different wavelengths. (c) The phase shift for incident light wavelengths as ne changes. (d) The solid curves present the sensitivities and the dashed curves show the corresponding loss properties. The heavy lines highlight the results from the dashed black boxes in (c).
Fig. 5
Fig. 5 (a) The phase difference curves for a range of different polishing depths. Different colors represent different polishing depths, while different marker types represent different values for the environmental refractive indices ne = 1.333 – 1.345. (b) The resonance wavelength λres(1.339) for different polishing depths. (c) The phase shift curves as ne increases. (d) The sensitivity for different polishing depths and the fitted result.
Fig. 6
Fig. 6 Fiber structural parameters and their influence in sensor performance. The left side displays the impact of the pitch on the sensitivity, and the right side for the cladding rod radius. The structural variants are plotted in different color, while different marker types represent different ne. The top row is the phase difference curves, the second row shows the phase shift for different ne. The sensitivities are plotted in the third row. The fourth and bottom rows present the electric field along the y-axis and the corresponding fitted results. The red circle in (g) highlight the protrusion of the wave vector under small pitch condition Λ = 1.4μm.
Fig. 7
Fig. 7 (a) The cross section of D-shaped step fiber sensor. The environmental refractive index is ne = 1.339 for comparison. (b) The evanescent field along the y-axis. The solid line represents fiber sensors based on PCF, while the dashed line represents fiber sensor based on step fiber. Different color corresponds to different depths. (c) Comparison of the maximum surface field (logarithmic axes). (d) The corresponding phase difference curves and sensitivities for ne = 1.339 RIU and polishing depth h = 1.1 Λ.

Equations (7)

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E ( r , t ) = Re ( E ˜ ( r ) e ζ z + i ω t ) = Re ( E ˜ ( r ) e k 0 Im ( n eff ) z i k 0 Re ( n eff ) z + i ω t
I out = E E * = I p + I s + 2 I p I s cos ( ω 0 t + θ ) I D C + 2 ( 1 2 k p I p 0 ) ( 1 2 k s I s 0 ) cos ( ω t + θ ) I D C + k p I 0 cos ( ω t + θ )
Φ d = 2 π λ ( Re ( n p ) Re ( n s ) ) L
S = a 1 exp ( b 1 x ) E max = a 2 exp ( b 2 x )
S = a 2 a 1 ( b 1 / b 2 ) E max ( b 1 / b 2 )
Γ = E d S Γ ( E max )
structural variable Γ ( E max ) sensitivity

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