## Abstract

We report a simplified model for the computation of light-fluorescence interactions within photonic crystal fibers (PCFs). It involved the plotting of ray trajectories confined by total internal reflection within a geometrically simplified PCF core. This was followed by the calculation of absorption and fluorescence emission at each point of reflection, which were subsequently summed and averaged over all the launched rays. The computation of these components for two specified wavelengths (peak excitation and emission) produced a dimensionless ratiometric relationship for varying concentrations of fluorescence dye. This hence eliminated the need for optical filters and minimized the effects of intensity fluctuations. Modeled results were demonstrated to concur well with that obtained experimentally for two PCFs with different microstructured cores.

© 2014 Optical Society of America

## 1. Introduction

The devise of photonic crystal fibers (PCFs) [1, 2] has opened a myriad of new avenues in the field of fiber optic technology. In particular, their periodic arrays of laterally-running microstructured air holes serve not only to confine and guide light via novel means, but further as fillable channels for in-fiber light-matter interactions – seeing to their numerous applications [3–6]. Index guiding PCFs, a class of PCF that guides light via modified total internal reflection (TIR) [7], offer broadband guidance and surface-specific light-matter interactions. The latter being manipulable by microstructure variations [8], has allowed for their applications in absorption spectroscopy of liquids [9–11] and gases [12, 13], as well as fluorescence spectroscopy [14, 15].

Often in the above employment of PCFs, the interfacing of optical fiber components to conventional optics involves bulky and component-intensive set-ups that largely overshadow the compactness and flexibility offered by fiber optics. In fluorospectroscopy, specifically, there is the further requirement of optical filters and/or laser sources. These serve to remove or allow for easy disregard of the remaining excitation, but are however extremely space consuming and often costly. Although several fluorospectroscopy approaches circumvent the need for traditional or physical optical filters, they involve using complex micro architectures that filter [16] or are spectrally selective detectors [16–20]; require highly precise detectors for fluorescent lifetime measurements [21–23]; or entail the use of an additional reference emitter for ratiometry [24].

Hence, to maintain the simplicity and low-cost of our previously reported all-fiber opto-fluidic platform [25], a mathematical spectral decomposition approach was employed. In brief, it characterizes the ratiometric change between the collected intensities at two specified wavelengths upon fluorescence dye concentration variations. This contrasts their respective remaining excitation and produced fluorescence emission, and provides a dimensionless relationship that mitigates the undesirable effects of intensity fluctuations. Since the region of light confinement and propagation in PCFs, namely its core, possesses dimensions relatively larger than the employed wavelength [26, 27], a ray-tracing approach was deemed a suitable approximation to model the in-fiber light-matter interactions. This is in general a less computationally intensive approach as compared to mode solutions computed from Maxwell’s equations employed in fundamentally similar index guiding microstructured optical fibers, used in fluorospectroscopy [14]. On the other hand, the model also complements the cuvette-like fluorospectroscopic methods performed in photonic bandgap guiding PCFs [15].

This paper focuses on the development of a ray-tracing model capable of accounting for the constituents – remaining excitation and produced emission – of the dimensionless ratiometric change over various dye concentrations. This was subsequently employed in the calculations for two PCFs, namely a defected-core PCF (dcPCF) [10] and a solid-core PCF (scPCF) [28]. These were then compared with experimentally obtained results for the corresponding PCFs loaded with varying concentrations of a common fluorescence dye – carboxyfluorescein.

## 2. Materials and methods

#### 2.1 Chemicals and other materials

5(6)-Carboxyfluorescein (CF) was purchased from Sigma-Aldrich (Singapore). 1M Tris buffer at pH8.8 was obtained from 1st Base (Singapore). All other chemicals were of reagent grade.

#### 2.2 Optofluidic platform set-up

The set-up, as shown in Fig. 1, comprises: (1) a blue (490nm) fiber-coupled LED (Thorlabs M490F1) as the excitation input; (2) a spectrometer (Ocean Maya 2000) for the spectral output collection; (3) a series of optical fibers (input & output multi-mode fibers and a length of PCF as the sensing element), fiber connectors and adapters; (4) a syringe-pump-attached opto-fluidic manipulator described earlier in [25]. The 105mm length of PCF is bent into a U-shape with a semi-circle segment, of bending radius (R_{bend}) 12.5mm, flanked by two equal straight segments.

A series of CF solutions, dissolved to various concentrations (10-100µM) in 321mM Tris buffer, was infiltrated into the bent of length PCF with their resultant output spectra collected subsequently. The ratio between the intensities at λ_{exc} = 490nm and λ_{ems} = 510nm (I(λ_{ems})/I(λ_{exc})) was then recorded per concentration.

#### 2.3 Ray-tracing model

To support the proposed spectral interpretation, a ray-tracing model describing fluorescence excitation and collection at the exposed surface of a conventional optical fiber [29] was adapted. Each ray, defined by its initial spatial location and angle of launch, has its trajectory and incident angles traced via simple geometrical constraints. Using the light-fluorescence interactions at each point of incidence and its number of occurrences, the spectral variations per concentration of CF, I(λ_{ems})/I(λ_{exc}) in particular, can thus be derived.

The launch condition per ray was determined by a combination of parameters [illustrated in Fig. 2(b) and Fig. 2(c)], namely its (1) x- and y-position of launch (xPos and yPos respectively); (2) angle of launch (θ_{START}), on its plane of launch; (3) tilt (ψ) of its plane of launch. This is with the assumption that the core of the dcPCF, of diameter D_{core}, can be approximated as a cylinder with a hollowed-core, of diameter D_{def-core}. The aforementioned core is defined as the central region surrounded by the innermost ring of air holes as indicated in Fig. 2(a). The same approximation was made for the scPCF, however, with its D_{def-core} ignored.

In order to incorporate the effect of bending, a graded core refractive index (n_{co}) profile was adopted. Bending is assumed here to have negligible effect on the medium refractive index (n_{m}), which is fluid in nature. The bending-induced refractive index change was governed by the following equation [30, 31]:

_{bend}. It should be noted that bending only alters n

_{co}along the x-axis, n

_{co}is hence a constant in the y- and z- direction. Here, the layer thickness (t

_{Lay}) – equivalent to D

_{core}divided by the total number of layers (Lay

_{Tot}) – served to provide a discretization to the n

_{co,bent}distribution as well as defined the mesh density of the x-y grid. This meant a density equivalent to Lay

_{Tot}

^{2}, where each x-y coordinated corresponded to a point of launch as indicated by the intersecting dotted lines in Fig. 2(b). Subsequently, from each launched ray, a set of geometrically derived values was obtained: (1) number of reflections (N

_{Refl}); (2) n

_{co,bent}at point of incidence; (3) incident angle (α).

A launched ray traverses the discretized layers undergoing refraction each time it crosses a layer-layer interface, bending away from the normal as it propagates in the positive x-direction – where n_{co,bent} decreases. Refraction continues until the ray experiences TIR, particularly when it meets the medium of a significantly lower n, or in unique scenarios between n_{co,bent} layers, when θ_{START} tends towards 90°, as depicted by the orange coloured ray in Fig. 3. Based on this, α is obtained for reflections off the top and bottom interfaces, α_{Top} and α_{Btm} respectively, or just α_{Btm} for the latter mentioned intra-core reflection. Concomitantly, n_{co} of the layer where reflections occurred were recorded. Next, using the longitudinal distance between TIR events, defined as L_{z} in Fig. 3, N_{Refl} can be calculated from the total length of PCF (L_{PCF}) under study.

Now, transforming the earlier discussed ray-trace into a 3D format (when ψ≠0), where each layer is converted into a cuboid element as illustrated in Fig. 4 (detailed derivations as well as rotated perspectives can be found in the Appendix). Drawing further geometrical relationships and employing Snell’s Law, the following is derived for the refraction regime shown in Fig. 4(a):

Consequently, the longitudinal displacement of the refracted ray is:

Similarly, the same was performed for the reflection regime, shown in Fig. 4(b), and the incident angle is given as:

Reflection in the plane of incidence, at the Top core-medium interface, gives an incident angle α_{Top} = θ_{2}’, which is also equivalent to the reflected angle θ_{3}’. It should also be highlighted here that reflection is assumed to occur along the edge of the cuboid, which defines the plane of incidence. Likewise, the above can be rewritten to accommodate rays launched at ψ>90°.

Generally, the emission of fluorescence presents as a shoulder on the right flank of the remaining excitation source, as depicted by the red-dotted spectrum in Fig. 5. This stems from the fact that the spectrum of fluorescence emission overlaps that of the remaining excitation, essentially implying that I(λ) is a superposition of the two. Employing that derived above with the assumption of negligible dispersion between the wavelengths in the studied span, the specific extent of this light-fluorescence interaction could then be calculated. These could subsequently be fitted into the following equation describing I(λ_{ems})/I(λ_{exc}), where λ_{exc} = 490nm and λ_{ems} = 510nm correspond to the peak absorption and emission wavelengths respectively. Its dimensionless form eliminates the common but undesirable effects of intensity fluctuations – typical of optical set-ups.

Considering the earlier mentioned spectral superposition, I(λ) was decomposed into its various constituents – I_{Fl}: Intensity of fluorescence emission; I_{0}: Initial (or reference) intensity; I_{abs}: Intensity absorbed – with the assumption of negligible reabsorption at low CF concentrations. (I_{0}-I_{abs}) being a representation of the remaining excitation. In addition, to simplify subsequent calculations, its components were normalized to I_{0}(λ_{exc}) and a fraction of absorption (f_{abs}), relative to I_{0}(λ)_{,} was introduced.

The above would be calculated for each launched ray and averaged, which effectively involved a 4D matrix (xPos by θ_{START} by yPos by ψ) representing all the ray launch conditions. Consequently, I_{Fl,norm} and f_{abs} were calculated by adapting the method described in [29], and are detailed as follows.

I_{Fl,norm} comprises the total fluorescence emission generated per launched ray. This would be the sum of emission resulting from all reflections off the core-medium interfaces, where each of these emissions is given by:

_{0}(λ)/I

_{0}(λ

_{exc}). c, ε and Φ

_{Fl}represent the concentration, extinction coefficient and quantum yield of the fluorescence dye, correspondingly. f

_{Fl,λ}is the fraction of emission at λ

_{ems}or λ

_{exc}. Lastly, F

_{abs}and P

_{emit}is the factor of absorption and the probability of emission collection respectively. These two terms are integrated over δ, which is the perpendicular distance from the core-medium interface extended infinitely into the medium.

The factor of absorption, F_{abs}, is defined by:

_{abs}is the power absorbed per unit volume and n

_{rel}is the relative refractive index (n

_{co,bent}/n

_{m}), while the penetration depth, d

_{p}, is:It should be highlighted that in all calculations of n

_{rel}, n

_{m}is assumed to be a constant equivalent to that of water, due to the negligible influence that low concentrations of solutes (fluorescence dye in this case) has on refractive index.

On the other hand, the probability of emission collection, P_{emit}, which discriminates for forward propagating (positive z-direction) rays:

_{#}/dΩ is the dipole radiation near a dielectric interface (the core-medium interface in this case) per unit solid angle (Ω), averaged over all dipole orientations. The subscript “#” corresponds to the different ranges of α between 0 and π, and are described in the Appendix.

From F_{abs}, the fraction of absorption, f_{abs}, was derived as [32]:

As mentioned earlier, each launched ray experiences top and/or bottom reflections off the core-medium interface. This therefore leads to a distinct value for each interfacial reflection – S_{ray,Top} and S_{ray,Btm} as well as f_{abs,Top} and f_{abs,Btm}, corresponding to their respective α and n_{co} values.

Further to which, as a fraction of I_{0} is absorbed after each reflection, I_{Fl,norm} is not a mere multiplication of S_{ray} by its total number of occurrence (N_{Refl}). Instead, it is expressed in the following two summations derived from the geometric progression of successive absorptions, one for top-and-bottom reflections (ΣS_{ray,TopBtm}) and the other for intra-core reflections, where the latter involves bottom-only reflections (ΣS_{ray,BtmONLY}).

Summing the above two equations for all λ gives the I_{Fl,norm} for each launched ray gives:

Similarly, the following two equations account for the fraction of remaining input intensity after absorption (1-f_{abs}) for λ_{ems} and λ_{exc}.

It should be highlighted here that Eq. (6) to (15) were only valid for launched rays capable of propagation through the entire L_{PCF}, i.e. underwent TIR. For rays that failed to meet the requirements for TIR or were, in other words, refracted, there was no remaining input excitation. This reduced I(λ_{ems})/I(λ_{exc}) to a ratio between the emission at λ_{ems} and λ_{exc} as shown below.

The above discussed ray-tracing model was translated into an algorithm in MATLAB and employed in the calculations for the following definitions and launch parameters, which were selected to closely match experimental conditions – D_{core} = 10µm and D_{def-core} = 4.1µm; R_{bend} = 12.5mm and L_{PCF} = R_{bend}*π; n_{co} = 1.4629 and n_{m} = 1.3373 (based on n at λ = 490nm), hence θ_{Crit} = sin^{−1}(n_{m}/n_{co}) = 66.1°; λ = 400-600nm; Lay_{Tot} = 101 and t_{Lay} = D_{core}/Lay_{Tot}; xPos = 0 to D_{core} at intervals of t_{Lay}; yPos = 0 and t_{Lay}/2 to (D_{core}/2-t_{Lay}) at intervals of t_{Lay}; θ_{START} = θ_{Crit} to (θ_{Crit} + 0.9°) at intervals of 0.1°, (θ_{Crit} + 1°) to 88° at intervals of 1° and 88° to 89.9° at intervals of 0.1°; ψ = 0° to 170° at intervals of 10°. It should be noted that yPos and ψ were only calculated for half the true span due to symmetry about the xPos axis. The resulting I(λ_{ems})/I(λ_{exc}) values were subsequently weighted according to their respective intervals and averaged for the CF concentrations of 0 to 100µM at 10µM intervals. CF parameters – ε(λ) was obtained experimentally; Φ_{Fl} = 0.93 [33, 34]; f_{Fl,λems} = 0.0175 and f_{Fl,λexc} = 0.00154 (from Life Technologies’ online database). This was likewise done for a scPCF with D_{core} = 13.8µm and D_{def-core} = 0µm.

## 3. Results and discussion

The calculated I(λ_{ems})/I(λ_{exc}) values from the ray-trace model were contrasted with those obtained experimentally. This was similarly done for an unbent or straight length of PCF (i.e. R_{bend} = ∞). However, based on the numerical aperture calculation described in [35], the θ_{START} range for the straight dcPCF was shortened to θ_{START} = (θ_{Crit} + 19) to 88 at intervals of 1 and 88 to 89.9 at intervals of 0.1, which more closely concurs with single-mode operation. On the other hand, θ_{START} = (θ_{Crit} + 21) to 88 at intervals of 1 and 88 to 89.9 at intervals of 0.1 for the straight scPCF.

The comparison in Fig. 6(a) showed good agreement between the theoretical and experimental results for the dcPCF. Contrasting the results between its bent and straight configurations, indicated the θ_{START} span as the primary reason for their distinction. Where smaller θ_{START} values (present only in the bent dcPCF calculations) resulted in smaller α, leading to larger S_{ray} and f_{abs} values, as described by Eq. (6) and (10) respectively.

Additionally, a smaller α would essentially imply more frequent reflections and therefore a larger N_{Refl}. This effectively dominates in I_{Fl,norm} (Eq. (11) and (12)) and 1-f_{abs} (Eq. (14) and (15)), both scaling to the power of N_{Refl}. Specifically, the former is large while the latter is small at smaller θ_{START} values, as shown in Fig. 7(a). Note that the jump in N_{Refl} at θ_{START} values close to θ_{Crit} was a result of launched rays being refracted; hence N_{Refl} is 0 for such rays. On the other hand, although CF concentrations linearly scale S_{ray} and f_{abs} values, the eventual calculations of I_{Fl,norm} and 1-f_{abs} were still dominated by N_{Refl}. Lastly, the difference between the 1-f_{abs} plots for λ_{ems} (1-f_{abs,ems}) and λ_{exc} (1-f_{abs,exc}) was due to a larger ε for the latter, which hence led to lesser remaining excitation. With reference to Eq. (5), a more rapidly decreasing 1-f_{abs,exc} with increasing CF concentrations and the inherently smaller I_{Fl,norm}(λ_{exc}), causes I(λ_{ems})/I(λ_{exc}) to increase in the observed exponential manner.

Next in Fig. 6(b), the scPCF plots for the bent configuration, were observed to only be agreeable upon adjustment of its θ_{START} span. This involved shortening the span and commencing it from 67.1°, instead of the original 66.1° (θ_{Crit}). Interestingly, back calculating n_{m} using 67.1° as θ_{Crit}, gave a larger refractive index of 1.3474. This larger n_{m} could however be understood as the effective refractive index of the dye-filled air holes surrounding the core, which upon further inspection of the scPCF’s cross section does concur with the more significant presence of the fused silica material between said air holes.

Contrasting between the bent PCFs, highlights two advantages of dcPCFs: (1) higher sensitivity and (2) higher repeatability, particularly at lower CF concentrations. These could primarily be attributed to its much larger N_{Refl} values, which as shown in Fig. 7 and discussed earlier results in larger I_{Fl,norm} and smaller 1-f_{abs} values. Furthermore, the lower repeatability experienced in the scPCF experiments, were also deduced to be a result of its lower radial homogeneity due to the earlier mentioned significance of fused silica. This implied that upon bending, the x’-axis had a much higher probability of crossing a fused silica segment (between two air holes), instead of a dye-filled air hole. For the same reason, it extends to explain why the ray-tracing model produced a less coherent fit for the scPCF. Nonetheless, the difference between the bent and straight configurations was still identical for both PCFs.

Notably, the above discussed calculations only required the experimental inputs of a corresponding reference spectrum, I_{0}(λ)/I_{0}(λ_{exc}), and figures describing CF’s optical properties – ε, f_{Fl} and Φ_{Fl}.

## 4. Conclusion

A ray-tracing model was demonstrated to be effective in the computation of in-fiber light-matter interactions in index guiding PCFs – a dcPCF and a scPCF. This was attained via the simplification of said PCFs’ core into cylinders, and plotting the trajectory of rays propagating via TIR along its length. The light-matter (light-fluorescence) interactions experienced at each point of reflection were further accounted for through calculations of absorption and collectable fluorescence emission. Subsequently, these interactions were summed over their total number of occurrences and averaged over all the launched rays. Combining these components produced the dimensionless ratio, I(λ_{ems})/I(λ_{exc}), which served in the experiment to not only mitigate the effects of intensity fluctuations but also to exclude the need for optical filters. Comparing the experimental and calculated values validated the described model – particularly for the dcPCF. However, this was also limited by the accuracy of the earlier made cylindrical approximations, as shown by the observed discrepancies for the scPCF whose core possessed a less continuous perimeter of medium.

## Appendix

The following lists the derivation for the symbols indicated in Fig. 4 and Fig. 8, for the refraction (A to E) and reflection (F to J) regime.

On the other hand, the following accounts for the different ranges of α described in Eq. (9), where 1: 0≤α≤α_{crit}, 2: α_{crit}≤α≤π/2 and 3: π/2≤α≤ π.

*const*is a constant that can be ignored in Eq. (9).

## Acknowledgment

This work is supported by the Agency for Science Technology and Research through the Advanced Optics in Engineering Programme and Graduate Scholarship. We would also like to thank Yangtze Optical Fibre and Cable Company Ltd. (YOFC) for assisting with the fabrication of the PCF.

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