## Abstract

We investigate theoretically the performance of photonic crystal fibres with coated holes as refractive index sensors. We show that coating the holes with a high-index material allows to extend the extreme sensitivities analyte-waveguide based geometries offer to the case of low-index analytes, including water-based solutions. As the sensitivity of these sensors is intricately linked to the sensitivity of the cutoff of a single inclusion to the analyte refractive index, our approach relies on the derivation of cutoff equations for coated inclusions. This is performed analytically without approximations, in the fully vectorial case, for modes of all orders. Our analytic approach allows us to rapidly cover the parameter space, and to quickly identify promising geometries. The best results are obtained when considering fluorinated polymer fibres, for which the index of the background material is not too different to that of water, and with thin high-index coatings. Using these results, we propose a sensor based on a directional coupler geometry that would lead to a sensitivity of 2.2×10^{4} nm=RIU for water based solutions with achievable smallest detectable refractive index changes below 10^{-6}.

©2009 Optical Society of America

## 1. Introduction

Photonic crystal fibres (PCFs) are optical fibres incorporating microscopic holes running along their length. As these holes are close to the optical core and can be filled with fluids, they provide an interesting platform for fibre based microfluidic optical sensing [1], be it for measurements of a fluid’s refractive index, temperature, fluorescence signals or biochemical agent concentrations.

Approaches for sensing properties of fluids within holes can be subdivided into two categories: the first comprises techniques based on absorption [2], fluorescence [3] or Raman scattering [4, 5] and relies on optical properties of the molecules to be detected. In most of these schemes, light is propagated in the PCF’s core, and it is the evanescent tail of the light in the PCF’s holes that probes the optical properties of an analyte. The benefits of using PCFs rather than equivalent techniques based on cuvettes and bulk optics lie in the long interaction lengths with small sample volumes that one can achieve, and the potentially very low cost of mass production of optical fibres.

The second category of PCF based sensors relies on resonant features of a PCF that can be very sensitive to the refractive index within the holes. Such resonant features can be either due to Bragg or long period gratings (LPGs) [6–8], to intermodal interferences, e.g. in tapers [9], to surface plasmon resonances [10, 11], to photonic bandgap properties of the PCF themselves [12, 13], or to resonances in a directional coupler geometry [14]. In the latter two geometries (Figs. 1 and 2) the analyte fills one or several of the PCF’s holes, and it is the optical resonances of these fluid channels which are probed. Since the analyte is part of the probed resonator, overlap with light is very high and remarkable sensitivities exceeding 3×10^{4} nm/refractive index unit (RIU) and detection limits below 10^{-6} RIU can be achieved [14–16]. However, the analyte channel only forms a resonator if it has a refractive index higher than that of the surrounding background material. Effectively the analyte channels form waveguides, and it is the coupling from the PCF’s core to the modes (resonances) of the analyte waveguides that is probed [17].

In many cases of interest, and in particular for biological and medical applications, analytes are water based solutions, with a refractive index of 1.33–1.35. Glasses and polymers suitable for drawing optical fibres on the contrary have much higher refractive indices, e.g. 1.45 for fused silica or 1.49 for Poly(methyl-methacrylate) (PMMA). While in principle optical fibres can also be drawn using fluorinated polymers with lower refractive indices, many of these still have a refractive index of order 1.38 [18].

As water based analytes have lower refractive index than all available materials for PCF fabrication, it appears it is impossible to use the above mentioned approaches based on the analyte forming a waveguide within the PCF. It has been suggested that adding a high-index coating to the holes of the PCF may be a way to circumvent this issue, and achieve the extreme sensitivities analyte waveguide based geometries offer even with low-index analytes [19]. Indeed, high-index rings still act as resonators even when they are filled with low index analytes, and PCFs having low-index filled high-index rings in an intermediate index background have been shown to be bandgap guiding [20]. How far a high-index coating enables to extend analyte waveguide based sensing schemes to low index fluids, however, has never been investigated in detail.

We note that a similar approach has received some attention in the recent literature, consisting in using metallic rather than dielectric coatings of holes. The metal/analyte interface supports surface plasmon resonance (SPR) modes, to which light from the core can couple to [10,11,20]. Such plasmonic fibre sensors are a promising, alternative approach for low-index PCF based refractive index sensing. Both approaches in principle lead to comparable sensitivities and detection limits. However to get narrow SPR peaks, metallic surfaces of tens of nm-thickness with nm-level smoothness are required, which are hard to achieve in a fibre [4, 21]. Dielectric coatings have the benefit of being thicker and more tolerant to low level surface roughness, as the waveguide modes used for sensing are bulk rather than surface modes. In the best case scenario the coating can be incorporated in the preform and co-drawn, so that no post-processing of the fibre is required (this is for example the case with polycarbonate and PMMA). Alternatively, high-index materials such as TiO_{2} can be deposited using SOL-GEL techniques by flushing the reactants through the holes [22].

Here we develop a generic framework to study the sensitivity of sensing approaches based on analyte waveguides in PCFs with holes coated by dielectrics. Our approach relies on the study of cutoffs of single coated inclusions, as the sensitivity of both solid core photonic bandgap fibres (SC-PBGFs) based sensors and directional coupler sensors such as that described by Wu et al [14] are intricately linked to the sensitivity of a single inclusion’s cutoff (Figs. 1 and 2). First, we briefly discuss the link between cutoffs and the sensitivity. Then we derive cutoff equations for coated inclusions, in the general case. Finally, we use this formalism to cover a large part of practicable parameter space before concluding on the prospects of coated-waveguide based refractive index sensing schemes.

## 2. Cutoffs and Sensitivity

#### 2.1. Sensitivity and detection limit

White et al. have discussed in detail the link between sensitivity, detection limit and other sensor parameters [23]. A refractive index sensor typically has a resonant feature of spectral width Δ*λ* at a wavelength *λ*
_{r} depending on the refractive index of the analyte *n*
_{a}. The sensitivity is defined as

In many cases, the sensitivity can be derived using first order perturbation theory, leading to S being proportional to the overlap of the field with the analyte, and to a factor typically involving the inverse of the difference of group indices of two modes [8] which depends on the specific implementation of the sensor. Typically, to analyse the sensitivity of a whole class of sensors, a large number of situations and overlap integrals need thus be numerically calculated.

The detection limit *δ*
_{n0} is defined as the smallest detectable change in refractive index, and is typically the quantity of interest. White et al have shown that, assuming temperature noise is correctly compensated for, and assuming the width of the resonance Δ*λ* is much larger than the resolution of the spectrum analyser (which is typically the case for the geometries of interest here),

where SNR is the signal to noise ratio of the measurement in linear units. In the present article we study an entire class of sensors, for which Δ*λ* can greatly vary, being largely dependent on details of the implementation such as long period grating length in a SC-PBGF, or coupling constants and length in a directional coupler geometry. The sensitivity S however is to first approximation a function of the geometry of the single coated inclusion. We thus concentrate on the study of *S*, providing examples of implementations of actual sensors and resulting detection limits. For this study, we introduce the scaled sensitivity σ defined as

which has the benefit of being scale invariant, and note that if a resonance has quality factor *Q* the detection limit Eq. (2) becomes proportional to 1=(*Qσ*). To lower the detection limit, one has thus to build a system combining a narrow resonance (high Q) with large scaled sensitivity *σ*.

The two classes of analyte waveguide based PCF sensors we discuss are a) sensors relying on the shift of SC-PBGF’s bands [12, 13], including long period grating sensors in SC-PBGFs (Fig. 1) [15, 16], and b) directional coupler in PCFs (Fig. 2) [14].

#### 2.2. Solid core photonic bandgap sensors

Sensors based on SC-PBGF typically consist of solid core PCFs with an array of holes that are filled with an analyte (Fig. 1). The analyte has a refractive index that is higher than that of the background, and light is guided in the solid core by photonic bandgap effects. Using a simple antiresonant reflecting optical waveguide (ARROW) model [24], or by arguing that the bands form through the coupling of modes of the individual high-index analyte channel waveguides [25], one can show that the transmission windows of SC-PBGFs are delimited by the cutoff frequencies of LP_{0q} and LP_{1q} modes of the individual waveguides formed by the high-index analyte channels [24]. The cutoffs of the modes of the individual channels occur at the well known fixed normalised frequencies [26]

where *n*
_{a} is the analyte’s index, *n*
_{bg} the fibre’s background index, *ρ* the radius of the PCF’s holes, *λ* the vacuum wavelength at cutoff, and *V _{c}*=2.405, 3.832, … [26]. When the refractive index of the analyte

*n*

_{a}changes, so does the cutoff wavelength, so that the transmission spectrum of the entire SC-PBGF shifts in wavelength. It is this shift that then is used for sensing the refractive index of the analyte. Experimentally, the shift can be determined by directly measuring the shift of the edges of bandgaps, that is the edges of high transmission regions of the SC-PBGF. However, the edges of bands are ill defined, spectrally wide features, leading to large Δ

*λ*and poor detection limits. To improve the detection limit, spectrally narrower features can be introduced in the SC-PBGF’s transmission spectrum using for example long period gratings [15, 16] or avoided crossings with higher order bands [13]. To first order, these narrow spectral features simply follow the bands of the SC-PBGF when

*n*

_{a}changes, and the sensitivity of the cutoff wavelength to

*n*

_{a}can be used as a proxy to the sensitivity of all these resonant features. We will see that the sensitivity obtained from this cutoff analysis typically gives a very good approximation of the sensitivity of actual implementations, even though a more geometry-specific analysis, e.g. using perturbation theory and overlap integrals or full electromagnetic simulations cannot be avoided to obtain the final sensitivity of a given sensor accurately. However, the cutoff analysis allows us to rapidly cover a large part of parameter space, enabling us to quickly identify promising geometries.

In the case where inclusions of the SC-PBGF are simple circular holes filled with high-index analyte, the scaled sensitivity becomes [derived by taking the derivative of Eq. (4)]

In particular the scaled sensitivity diverges for analytes having refractive indices close to but larger than the background index. In combination with narrow resonances achieved by long period gratings, this can provide detection limits of order 10^{-6} [16]. When *n*
_{a}<*n*
_{bg}, the SC-PBGF becomes index-guiding, there are no cutoff-delimited transmission bands, and the above analysis is no longer valid. However, by coating the holes with a dielectric material of index *n*
_{b}>*n*
_{bg}, bandgap guidance can be restored, and again the transmission bands of the resulting SC-PBGF are delimited by the cutoffs of the coated inclusions [20]. Eqs. (4) and (5) cannot be applied for coated analyte-filled inclusions, and cutoff equations in this case are considerably more complicated. One can however imagine that if the “average” index of the coated inclusion is close to but higher than *n*
_{bg}, diverging sensitivities as those predicted by from Eq. (5) could be achieved. Our aim here is to see how far this is indeed true.

#### 2.3. Directional PCF coupler

Wu et al. suggested and demonstrated that the same large scaled sensitivity as for low index-contrast SC-PBGF based sensors, but with small spectral features and thus improved detection limit, can be achieved by filling a single hole with analyte rather than all holes of the cladding (Fig. 2) [14]. In that geometry, the PCF becomes a simple directional coupler, coupling light between the central core of the PCF and a satellite waveguide formed of an analyte-filled hole surrounded by air holes. As the effective index of modes of the central core is below *n*
_{bg}, while the effective indices of modes of the analyte channel are higher than that of the background material, satellite and core modes are never phase-matched and cannot couple. However, when a satellite mode reaches the cutoff wavelength, it expands outside the analyte channel and ends up being confined by the surrounding air holes. The satellite mode then takes effective index values below *n*
_{bg}. During this transition, coupling to the central core mode can occur, which materialises as a narrow dip in the core mode transmission curves. This coupling necessarily occurs near the analyte channel’s cutoff wavelengths, and the scaled sensitivity is also given by Eq. (5).

As with SC-PBGF based sensors, this geometry is not suitable for analytes with lower refractive index than the background material, as these do no support modes. However, by coating the hole with a high-index material the structure can again support modes with effective indices higher than that of the background, and the directional coupler geometry can be extended to low index analytes, with a sensitivity that can be derived from that of cutoffs of coated cylinders.

## 3. Cutoffs of coated cylinders

The cutoffs of circularly symmetric inclusions of arbitrary radial refractive index profile can be obtained through numerical and approximate methods [27–29], and profiles similar to the one of interest here have been studied using such techniques [30]. Many of these techniques use the scalar approximation, which also makes analytic derivation reasonably straightforward. Karadeniz et al. recently derived the full analytic equation to find the cutoff frequencies of TE and TM modes in the fully vectorial case, but used numerical techniques to look at higher order cutoffs [31]. Here, given that we are interested in potentially strong index contrast, we aim at finding the analytic expressions without approximations, in the fully vectorial case, for modes of all orders.

#### 3.1. Fields and modes

The geometry we consider is that of Fig. 3, where *n*
_{b}>*n*
_{c} and in most cases of interest to us *n*
_{b}>*n*
_{a}. The electric and magnetic modal fields *𝓔* and *𝓗* of such a structure can be expressed as

$$\U0001d4d7(r,\theta ,z,t)=\genfrac{}{}{0.1ex}{}{1}{{Z}_{0}}\mathbf{K}(r,\theta )\mathrm{exp}\left[i(\mathrm{\beta z}-\mathrm{\omega t})\right],$$

where *β* is the mode’s propagation constant, ω its angular frequency and Z_{0} the vacuum impedance. In each of the regions a, b and c, **E**(*r,θ*) and **K**(*r*,*θ*) satisfy the Helmholtz equation

where *k*
_{0}=ω/*c* is the wavenumber in vacuum, nj the refractive index in region *j*∈{a,b,c} and *ψ* is any component of either **E** or **K**. In cylindrical coordinates, the solution to the Helmholtz equation are the Bessel functions, and any field satisfying the Helmholtz equation in region *j* can thus be expressed as

where *J _{m}*(

*z*) and

*H*

^{(1)}

*(*

_{m}*z*) are the Bessel and Hankel functions of the first kind respectively, and ${k}_{\perp}^{j}=\sqrt{{n}_{j}^{2}{k}_{0}^{2}-{\beta}^{2}}$. We choose to express the fields in all regions

*j*in terms of these functions, letting k⊥ take imaginary values when necessary: the Bessel and Hankel functions of imaginary arguments are of equivalent use to the modified Bessel functions

*I*and

*K*of real arguments more traditionally used in the context of evanescent fields in circular geometries [32]. This choice of functions allows us to treat all cases of relative magnitudes of

*n*

_{a},

*n*

_{b},

*n*

_{c}without having to change basis functions and notations.

The terms in *A ^{c}_{m}* are standing waves sourced outside of the coated cylinder, while the terms in

*B*

^{a}

*have a singularity at the origin: for a mode all these coefficients are zero [33]. A mode is then a non-zero set of coefficients*

_{m}*A*

^{a}

*,*

_{m}*A*

^{b}

*,*

_{m}*B*

^{b}

*,*

_{m}*B*

^{c}

*that for a propagation constant*

_{m}*β*satisfies the boundary conditions at the interfaces ab (

*r*=

*r*

_{a}) and bc (

*r*=

*r*

_{b}). We define the normalised frequency as

and the effective index of these modes as *n*
_{eff}=*β*=*k _{0}*.

#### 3.2. Boundary conditions, scattering matrices, and modal equation

The boundary conditions are provided by the continuity of the *z* and *θ* components of **E** and **K**. Because of the circular symmetry of the boundary conditions, all orders *m* in Eq. (8) decouple and lead to separate modes of azimuthal order *m*. For each *m*, the boundary conditions at a single interface ab or bc can be expressed in terms of scattering matrices, linking the outgoing and incoming fields on both sides. Defining

we have at each interface

$${\mathbf{B}}_{m}^{+}+{\mathbf{S}}^{++}{\mathbf{A}}_{m}^{+}+{\mathbf{S}}^{-+}{\mathbf{B}}_{m}^{-},$$

where the + and - superscripts refer to the outside and inside of a single interface respectively. The scattering matrices for a single interface can be found analytically following, for example, the appendix of Ref. 33. They depend on the geometry and on w and neff. We are interested in the case *n*
_{b}>*n*
_{c}>*n*
_{a}: guided modes hence have *n*
_{b}>*n*
_{eff}>*n*
_{c} and are confined between the two interfaces ab and bc. At each interface, Eq. (11) is satisfied, with **A**
^{c}
* _{m}*=0 and

**B**

^{a}

*=0. We thus have*

_{m}Eliminating Bbm we have

which has non zero solutions only if

which, when det(S^{++}
_{ab})≠0 is equivalent to

Equation (16) is the modal equation we will use in the remainder of this article.

#### 3.3. Cutoffs

Solving Eq. (16) for *n*
_{eff} at a given *ω* allows us to find the modes of the structure. A mode is cut off when *n*
_{eff}=*n*
_{c}. In order to find the cutoff frequency of the modes, we need to take the limit of Eq. (16) for *k*
^{c}
_{⊥}→0, using the expansions of *J _{m}*(

*z*) and

*H*

^{(1)}

*(*

_{m}*z*) and their derivatives for small arguments [32], and solve in w. While this is a reasonably straightforward exercise when using the scalar approximation (where matrices in Eq. (16) are simple scalars), taking into account the fully vectorial nature of the fields leads to two major difficulties:

1. The small argument expansions for the Bessel and Hankel functions and their derivatives differ for different orders, leading to three separate cases with different nature of cutoff equations for *m*=0, *m*=1, and m≥2.

2. With increasing *m*, an increasing number of lowest order terms in the small argument expansions cancel out, and a large number of terms need to be kept in these expansions.

We note that in Eq. (16) only S--_{bc} depends on *k*
^{c}_{⊥}. Using Mathematica [34] to expand S--_{bc} in power series of *k*
^{c}_{⊥}, and after much simplification we find:

·*m*=0

· *m*=1

· *m*≥2

$$\left(EE\right)\genfrac{}{}{0.1ex}{}{-1}{{\delta}_{m}}\left[(m-1){r}_{\text{b}}{k}_{0}({n}_{\text{b}}^{2}{J}_{m}{H}_{m}^{\left(1\right)\prime}+{n}_{\text{c}}^{2}{H}_{m}^{\left(1\right)}{J}_{m}^{\prime})\sqrt{{n}_{\text{b}}^{2}-{n}_{\text{c}}^{2}}+{\tau}_{m}{J}_{m}{H}_{m}^{\left(1\right)}\right]$$

where

Here we introduced the condensed notation *J _{m}* and

*H*

^{(1)}

*m*and their 0 counterparts to represent the values of the Bessel functions and their derivatives on the inside of the bc interface,

*J*(

_{m}*k*

^{b}

_{⊥}

*r*

_{b}),

*H*

^{(1)}

*(*

_{m}*k*

^{b}⊥

*r*

_{b}), $d{J}_{m}\left(z\right)\u2044\mathrm{dz}{\mid}_{z={k}_{\perp}^{\text{b}}{r}_{\text{b}}}$, … Using these expressions in Eq. (16) provides the full cutoff equation. The only other matrix needed is (

**S**

^{++}

_{ab})

^{-1}; its coefficients are given explicitly in Appendix A. As usual, for

*m*=0 matrices are diagonal, the electric and magnetic fields decouple, and TE and TM solutions can be distinguished. In Eq. (18), for

*m*=1, the off-diagonal elements also vanish for

*k*

^{c}

_{⊥}→0; we have written out their lowest order dependence on

*k*

^{c}

_{⊥}explicitly as this will be useful for deriving the transverse decay length of these modes near cutoff in the next section.

#### 3.4. Numerical verification

Figure 4 shows an example of dispersion curves *n*
_{eff}(*V*) for the first two (TE and TM) *m*=0 modes as well as the first *m*=1 and *m*=2 modes, calculated using Eq. (16), along with their respective cutoff frequencies obtained from the limiting form of Eq. (16) for small *k*
^{c}
_{⊥}. For *m*=0 and *m*=2, agreement between the numerically calculated point at which *n*
_{eff}=*n*
_{c} and the cutoff frequency is perfect. The case for *m*=1 is less obvious: numerically it appears *n*
_{eff} reaches *n*
_{c} at higher frequencies than the cutoff frequency. In fact, from the off-diagonal elements of Eq. (18) one can show that *m*=1 modes have an asymptotic dependence near cutoff in

wherea is a positive constant and *V _{c}* is the normalised cutoff frequency of the considered mode. Such a function reaches values very close to

*n*even far from

_{c}*V*. We have been able to track the mode down to

_{c}*V*=1.001

*V*

_{c}, but to do so a working precision of several hundreds of digits was required, as the zeroes of Eq. (16) become extremely narrow. This emphasises how previously used purely numerical methods to find the cutoffs without the use of asymptotic expansions are unlikely to succeed for

*m*=1. It is worth noting that near the cutoff of

*m*=1 modes, the transverse decay length

*L*

_{D}=1=Im(

*k*

^{c}_{⊥}) of the fields in region c takes the form

which means that when getting close to *V*
_{c}, the modal field diameter expands exponentially into region c. In fact, in the case of the *m*=1 mode of Fig. 4, by the time *V*=1.01*V*
_{c} the mode is approximately 10 billion times larger than the coated hole, and exceeds the size of the universe for *V*=1.001*V*
_{c}. The exact cutoff frequency in that case becomes somewhat irrelevant, in particular for SC-PBGFs or directional couplers: for SC-PBGFs the field expansion means the modes of different inclusions become very strongly coupled, while in the directional coupler architecture it rapidly becomes a mode of the surrounding structure rather than of the coated inclusion. In such a case one should rather argue on the frequency at which the modal diameter reaches the limit of the closest external structure.

## 4. Results

Our aim is to map the scaled sensitivity σ for the sensing of water based solutions. We will use *n*
_{a}=1.35, a value typical of such solutions. We investigate two cases: first conventional high-index background materials, such as silica or standard optical polymers, using *n _{c}*=1.45; second a lower index background material such as Poly(heptafluorobutyl methacrylate) with

*n*

_{c}=1.383 [18].

#### 4.1. Silica background

Figure 5 shows σ and *V _{c}* of the first (TE) and second (TM)

*m*=0 modes as a function of

*r*

_{a}=

*r*

_{b}and

*n*

_{b}for

*n*

_{c}=1.45 and

*n*

_{a}=1.35. For both modes, s increases with

*r*

_{a}=

*r*

_{b}, which can be understood in that all things being equal the overlap of the mode with the analyte increases with increasing

*r*

_{a}. For large

*r*

_{a}=

*r*

_{b}, the scaled sensitivity increases rapidly with

*n*

_{b}before peaking at a value close to s

_{max}=4.1 for the first (TE)

*m*=0 mode, and s

_{max}=4.3 for the second (TM)

*m*=0 mode. In both cases, these correspond to diverging values of the cutoff frequency

*V*

_{c}. Best scaled sensitivity is thus achieved in situations in which the cutoff wavelength is smaller than the size of the coated hole, and for values of

*n*

_{b}⋍2. The maximum scaled sensitivity reached in this case would correspond to an absolute sensitivity of order 2.7×10

^{3}nm/RIU when using red light, or 6.4×10

^{3}nm/RIU when operating around 1.5

*µ*m. This is half an order of magnitude less than what has been demonstrated using high-index analytes [14], but is still sufficient for many applications. In particular, an implementation similar to that in Ref. [14] would yield a detection limit of 3×10

^{-6}RIU, similar to the best predicted results in other water based PCF sensing schemes, and over an order of magnitude better than that of any experimentally demonstrated PCF refractive index sensing scheme to date [8].

Figure 6 shows σ and *V*
_{c} of the first and second *m*=1 modes as a function of *r*
_{a}=*r*
_{b} and *n*
_{b} for *n*
_{c}=1.45 and *n*
_{a}=1.35. The scaled sensitivity and cutoff frequencies for the second *m*=1 mode follow closely those of the *m*=0 modes, with similar trends and σ_{max}⋍4.2. By contrast, the first *m*=1 mode behaves very differently. This mode is equivalent to the fundamental mode HE_{11} of step index fibres. While in step index fibres the fundamental mode is never cut off, for a coated cylinder the fundamental mode can have a finite cutoff frequency if its homogenised refractive index (in terms of averaged permittivities) is lower than the background index. The blue curve on the surface plots for this mode corresponds to *fn*
^{2}
_{a}+(1-*f*)*n*
^{2}
_{b}=*n*
^{2}
_{c}, with *f*=(*r*
_{a}=*r*
_{b})^{2} the filling fraction of *n*
_{a}. Beyond that curve the homogenised permittivity [35] of the inclusion is larger than the background permittivity *n*
^{2}
_{c}, and the mode is not cut off. Before the fundamental mode’s cutoff disappears, its scaled sensitivity diverges. However, the associated cutoff frequency goes to zero, meaning the cutoff wavelength by far exceeds the size of the coated hole: sensing schemes such as the directional coupler or SC-PBGFs then cannot use the mode near this homogenisation limit. It is hard to define a limit to the realistically achievable scaled sensitivity for this mode. In fact, and as pointed out above, because of their very fast expansion of modal fields near cutoff, the present cutoff analysis is of limited use to *m*=1 modes. However, simulations of a number of vastly different sensor implementations using *m*=1 modes and covering the range of Fig. 6 suggest the limit scaled sensitivity is similar to that of *m*=0 modes.

Overall, for the four modes discussed above, the scaled sensitivity is limited to values around 4, and maximum scaled sensitivity is achieved for lower order modes and highest *r*
_{a}=*r*
_{b} values. However, this also corresponds to large *V*
_{c} values. The high values of *V*
_{c} can correspond to short wavelengths compared to the hole size, which may not be practicable for actual implementation in the directional coupler geometry as described byWu et al [14] as in that geometry the hole size also determines the core geometry and core to hole distance (which in turn determines the coupling length). However, σ is a weaker function of *r*
_{a}=*r*
_{b} than *V*
_{c} for *r*
_{a}=*r*
_{b}>0.9, so that practicable *V*
_{c} can be used with little compromise on the scaled sensitivity. Furthermore, the directional coupler geometry does not need to be in a regular array of PCF holes, as discussed in the conclusion. In order to achieve even higher sensitivities for water based solutions, the background refractive index needs to be lowered.

#### 4.2. Fluorinated polymer background

We now study the case for *n*
_{c}=1.383, which corresponds to the index of Poly(heptafluorobutyl methacrylate). Figure 7 shows the scaled sensitivity and normalised cutoff-wavelengths for the first few modes with (*m*=0,1,2,3), for *r*
_{a}=*r*
_{b}=0.99, as a function of *n*
_{b}. Apart from the fundamental mode for which the cutoff analysis is only partly applicable, highest scaled sensitivity is once more achieved for the TM mode, and peaks for a value of *n*
_{b} between 1.5–1.7, with σ_{max}⋍13. This corresponds to an unscaled sensitivity of 2×10^{4} nm/RIU when using 1.5 *µ*m wavelength or 8×10^{3} nm/RIU when using red light.

We note that in an implementation as a directional coupler similar to that described in Ref. 14, σ=13 would provide a detection limit below 10^{-6}. It thus appears clearly that high sensitivity low-index refractive index sensing using SC-PBGF and PCF based directional couplers with high-index coated holes is possible. Optimal results will be achieved with a background index being as low as possible, and thin coatings (*r*
_{a}/*r*
_{b}>0.9), with refractive index optimised for the required sensitivity range.

## 5. Example

Based on Fig. 7, we have designed a directional coupler sensor for water-based solutions, *n*
_{a}=1.35, with parameters close to those achieving *σ*max. We have chosen a PCF with a structure similar to that shown in Fig. 2 based on a fluorinated polymer background, *n*
_{c}=1.383, with hole-to-hole distance 8 mm and coated hole radius parameters *r*
_{a}=1.92 mm and *r*
_{b}=2 mm. This gives a value of *r*
_{a}/*r*
_{b}=0.96 which is a compromise between realistic coating thickness, operation wavelength, and optimal scaled sensitivity. With that value, our cutoff analysis predicts a maximum scaled sensitivity of the TM (*m*=0) mode cutoff of one isolated coated cylinder of σ=11.16 obtained for a coating index of *n*
_{b}=1.5. The same parameters lead to a scaled sensitivity of σ=15.64 for the fundamental HE_{11}(*m*=1) mode. The normalised cutoff frequencies of these two modes are *V*
_{c}=15.2933 and 11.5647, respectively.

Using a multipole method for coated cylinders [20], we have then calculated the modes of the PCF’s core and the TM and HE_{11} modes of the analyte channel waveguide taking into account the entire sensor structure. The results are shown in Fig. 8 where we have plotted the effective indices *n*
_{eff,TM} of the TM and *n*
_{eff,HE} of the HE_{11} modes of the analyte channel versus wavelength. These calculations have been performed for two different values of *n*
_{a}, respectively *n*
_{a}=1.350 (solid blue curves in Fig. 8) and *n*
_{a}=1.351 (red dashed curves), so that we can then calculate the actual scaled sensitivity of the sensor by numerical differentiation. Figure 8 also shows the effective index *n*
_{eff,core} of the fundamental mode of the PCF’s core (defined by the missing hole in Fig. 2). In the directional coupler, transmission notches are measured at wavelength at which *n*
_{eff,core} and *n*
_{eff} of the modes of the analyte channel waveguide cross.

As seen in Fig. 8, the effective index of the TM mode crosses the background index *n*
_{c}=1.383 at a slightly shorter wavelength than the cutoff wavelength predicted by our analytic method (crosses in Fig. 8), which is expected since the holes surrounding the analyte waveguide start affecting the *n*
_{eff}-curves near cutoff. However, the scaled sensitivity calculated from the difference in wavelengths at which *n*
_{eff;TM}=*n*
_{c} for *n*
_{a}=1.350 and for *n*
_{a}=1.351 is σ=10.81, in very good agreement with the predicted value of 11.16. The scaled sensitivity obtained from the points at which *n*
_{eff,TM}=*n*
_{eff,core} is 12.1, the slightly higher value coming from the slope of *n*
_{eff,core}. This latter value is the scaled sensitivity that would be measured experimentally.

As expected, the HE_{11} mode’s effective index *n*
_{eff,HE} crosses *n*
_{c} much earlier than the analytically predicted cutoff wavelengths. This is because of the rapid transverse expansion of the fields of *m*=1 modes near cutoff. However, the scaled sensitivity calculated from the difference in wavelengths at which *n*
_{eff,HE}=*n*
_{c} for *n*
_{a}=1.350 and for *n*
_{a}=1.351 is 11.51, still in reasonable agreement with the analytically obtained value of 15.64. The scaled sensitivity obtained from *n*
_{eff,HE}=*n*
_{eff,core} is 14.5, or an absolute sensitivity of 8.8×10^{3} nm/RIU. Scaled for operation at 1550 nm as in Ref. [14], this corresponds to an absolute sensitivity of 2.2×10^{4} nm/RIU and, assuming similar resonance widths as in [14], a detection limit below 10^{-6}. We note that the exact resonance width can be modulated by design of the microstructure, but should be similar to that in Ref. [14].

## 6. Conclusion and discussion

Using an analytical analysis based on the modal cutoff of coated cylinders, we have investigated the theoretical sensitivity to water based solutions of a whole class of coated-waveguide based refractive index sensors. We have demonstrated that high-index coatings indeed make low-index sensing achievable. Sensors based on a silica background would have a maximum sensitivity of 2.7×10^{3} nm=RIU when using red light, or 6.4×10^{3} nm=RIU when operating around 1.5 *µ*m, with a detection limit that could be as low as 3×10^{-6} RIU. These figures are comparable to other PCF sensing schemes for water based solutions. Obtaining higher sensitivities require background materials with refractive indices closer to that of water, such as low-index fluorinated polymers. In particular, we have proposed a sensor design based on a directional coupler in a selectively-filled fluorinated polymer PCF. This sensor leads to an absolute sensitivity of 2.2×10^{4} nm=RIU around 1.5 *µ*m for water based solutions with achievable smallest detectable refractive index changes below 10^{-6} using picoliter sample sizes. Additionally, our comprehensive analysis reveals that the maximum sensitivities are always obtained with the lowest order modes, namely the TM and HE_{11} modes, which is due to their large overlap with the analyte. However, for biological sensing where sensitivity near the surface rather than in the bulk of the analyte is required, it may be beneficial to use higher order modes, so that a full characterisation of the surface sensitivity is still needed. Surface discrimination should also be possible using several modes at the same time.

While we have used the directional coupler geometry suggested byWu et al for our practical example, it should be noted that this implementation requires selective filling of the coated hole, which is hardly a viable solution outside the laboratory. This can be avoided by coating a single hole, and filling all the holes with the analyte. Light is still guided by total internal reflection in the core, and only the coated hole gives rise to a resonance. Similar geometries using a single coated hole and a single high-index region as a core can also be devised, and will be the topic of a future publication.

Finally, we should stress out the generality of our fully analytical vectorial cutoff analysis. In particular, it is independent of the relative magnitudes of the refractive indices of the different regions and could as well be applied to metallic coatings or coatings made of metamaterials.

## Appendix A: (S^{++}_{ab})^{-1}

Following a procedure similar to that described in Ref. 33 we find:

where

with the notations ${\alpha}_{{J}^{\pm}{H}^{\pm}}^{\pm}=({k}^{\pm}\u2044{k}_{0}){J\text{'}}^{\pm}{H}^{\pm},{J}^{\pm}={J}_{m}\left({k}^{\pm}{r}_{\text{a}}\right),{H}^{\pm}={H}_{m}^{\left(1\right)}\left({k}^{\pm}{r}_{\text{a}}\right),{k}^{-}={k}_{\perp}^{\text{a}},{k}^{+}={k}_{\perp}^{\text{b}},{n}_{-}={n}_{\text{a}},{n}_{+}={n}_{\text{b}}$, and with other a coefficients defined analogously. Note that the (*EK*) component corresponds to the 1st (*E*) line, 2nd (*K*) column of the matrix.

## Acknowledgements

The research of B. T. Kuhlmey and S. Mahmoodian was supported under the Australian Research Council’s (ARC) Discovery Project and Centre of Excellence funding schemes. CUDOS is an ARC Centre of Excellence. S. Coen acknowledges the support of the Foundation for Research, Science and Technology (FRST) of the New Zealand government. This paper is devoted to Anatol Coen without whose involuntary participation, this work would never have been completed.

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