Previous work [13
] demonstrated that the fundamental limitation to the sensitivity of fiber-based fluorescence sensors is the background fluorescence from the glass itself, which was discussed in detail in Section 2. To better understand the behaviour of these sensors, we chose to model the system using the analytic vectorial solutions of a simple step-index fiber, as opposed to a FEM model [21
], making the reasonable assumption that the supporting struts within the WW fiber have a negligible impact on the field distributions and fluorescence recapture. We define the effective core size as the diameter of the circle with an area equivalent to the largest triangle that fits wholly within the core [17
]. For the purposes of this modeling, the fiber loss was ignored, as since both the glass fluorescence and Qdot signal fluorescence are located in the same spectral region, they both experience a comparable reduction in signal under transmission. As such, the primary effect of loss on this system is a reduction in the measured signal to noise ratio.
We seek to gain an understanding of how the impact of the undesirable glass fluorescence on the sensor performance can be minimized. To do this, we define a figure of merit (FOM) for both the background glass fluorescence (FOMglass) and for the Qdot signal fluorescence (FOMsignal) as a measure of the intensity of the fluorescence that is captured into the backward propagating modes of the fiber. These FOMs give a measure of the total amount of fluorescence photons that are initially emitted and then recaptured in to the modes of the fiber. The former is a result of the amount of incident laser power on the fluorophores, and the latter due to the efficiency that photons emitted by the fluorophore are recaptured.
For the Qdot fluorescence, FOMsignal
is defined as the product of the power fraction (PF) of the fundamental mode at the excitation wavelength located within the holes of the fiber (PFclad
] and the fluorescence capture fraction (FCF) of the emitted fluorescent light back in to all guided modes of the fiber [19
Only the fundamental mode was considered for the excitation of the fluorescence, as through proper lens choice and careful alignment the laser light can be preferentially coupled into the fundamental mode of the fiber, even though the waveguide is capable of supporting higher order modes (HOMs) unless impractically small core sizes are considered.
For the fluorescence capture, it has been demonstrated that higher order modes must be included in the model, as the fluorescence capture into these modes has a significant impact on the total FCF, especially once the core size is increased well beyond the cut-off for single mode guidance [20
]. Further supporting this, the experimentally measured mode images ( below) show that while the excitation at 532 nm  is well confined to the core and displays no obvious HOM content, the fluorescence emission around 780 nm  clearly has substantial HOM content.
Figure 6. (a) Mode image of 532 nm excitation. Fiber filled with decane to ensure correct mode profile for comparison to fluorescence mode image. (b) Mode image of very high concentration (1 μM) Qdot 800 fluorescence mode excited with 532 nm source. Scales (more ...)
Similarly for the glass fluorescence, the same definitions and assumptions are made, with FOMsignal being defined as the product of the PF within the glass (i.e., 1 − PFclad) and the FCF of the glass emitted light captured by all backward travelling modes of the waveguide.
These quantities can now be brought together to provide a better understanding of the overall performance of the sensor. This is done by defining a figure of merit, FOMtotal = FOMsignal/FOMglass. The primary constraint currently limiting the minimum detectable concentration of the sensor is the amount of Qdot signal fluorescence relative to the glass fluorescence. FOMtotal, shown in , is a direct measure of the relative magnitude of detected Qdot signal relative to the glass that is observed by the sensor.
Figure 7. (a) Power fraction at the excitation wavelength located within the cladding or core for various glasses, decane filled. (b) Figure of Merit (FOM) for fluorescent photons excited within the core of the fiber, this represents the intrinsic glass fluorescence. (more ...)
As can be seen from , reducing the core size fiber has a significant impact on FOMtotal
(regardless of the glass choice), and thus on the performance of the sensor, with improving performance predicted for the smallest core sizes. However, it is important to note that loss mechanisms were not included in this study. It has been previously shown [17
] that the loss increases significantly due to scattering from surface roughness both in tapered nanowires and suspended nanowires once the core size becomes sufficiently small. Additionally, modes close to cutoff also experience significant confinement loss [20
], resulting in both a reduction in signal at small core sizes where FOMtotal
is predicted to be largest using the simple model derived here, and a reduction in signal at each point where an additional mode is guided as modes close to cutoff are generally spread out and not well confined.
The final parameter examined within this theoretical study was the effect of the index contrast between the glass and the cladding (). Three glasses were examined: F2, LLF1 and Tellurite. The first two are commercially available lead silicates, with the third an in-house fabricated glass, with refractive indices at 532 nm of 1.63, 1.55 and 2.02 respectively. The refractive index used for the decane within the holes was 1.41, giving Δn values of 0.22, 0.14 and 0.61 respectively.
As shown from the plots in reducing the index contrast results in an increase in FOMtotal which corresponds to an increase in the amount of Qdot signal fluorescence observed relative to the background glass levels. However when the intensity of the glass fluorescence for each of the three glasses (as shown in ) is considered the F2 or F2HT glasses will in practice show the lowest background glass signals and therefore the best sensitivity. This is a consequence of the F2HT fluorescence intensity being 3.3× lower than LLF1 at 780 nm, and 4.6× lower than the Tellurite fluorescence at the same wavelength. So even though the overall capture of the fluorescence photons is reduced for the LLF1 glass, since the efficiency with which they are generated within the glass is significantly higher the net result in fiber is a higher glass fluorescence signal. This is further exacerbated by using a higher index glass such as Tellurite, as more fluorescence is generated in the first place and since FOMtotal is significantly lower with the higher index glass this results in a very large glass fluorescence signal relative to the Qdot signal.
These results demonstrate that a reduction in the index contrast results in a decrease in FOMglass, and depending on the core size can also show an increase in FOMsignal leading to an overall increase in FOMtotal.