## Coated photonic bandgap fibres for low-index sensing applications: cutoff analysis

Optics Express, Vol. 17, Issue 18, pp. 16306-16321 (2009)

http://dx.doi.org/10.1364/OE.17.016306

Acrobat PDF (504 KB)

### 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

1. T. M. Monro, W. Belardi, K. Furusawa, J. C. Baggett, N. G. R. Broderick, and D. J. Richardson, “Sensing with microstructured optical fibres,” Meas. Sci. Technol. **12**, 854–858 (2001).
[CrossRef]

2. T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer, and P. J. Sadler, “Quantitative broadband chemical sensing in air-suspended solid-core fibers,” J. Appl. Phys. **103**, 103108/1–7 (2008).
[CrossRef]

4. A. Amezcua-Correa, A. C. Peacock, C. E. Finlayson, J. J. Baumberg, J. Yang, S. M. Howdle, and P. J. A. Sazio, “Surface enhanced Raman scattering using metal modified microstructured optical fibre substrates,” in *32nd European Conference on Optical Communication*, p. Tu4.3.4 (Cannes, France, 2006).
[CrossRef]

5. D. Pristinski and H. Du, “Solid-core photonic crystal fiber as a Raman spectroscopy platform with a silica core as an internal reference,” Opt. Lett. **31**, 3246–3248 (2006).
[CrossRef] [PubMed]

6. L. Rindorf, J. B. Jensen, M. Dufva, L. H. Pedersen, P. E. Hoiby, and O. Bang, “Photonic crystal fiber long-period gratings for biochemical sensing,” Opt. Express **14**, 8224–8231 (2006).
[CrossRef] [PubMed]

8. L. Rindorf and O. Bang, “Sensitivity of photonic crystal fiber grating sensors: biosensing, refractive index, strain, and temperature sensing,” J. Opt. Soc. Am. B **25**, 310–324 (2008).
[CrossRef]

9. D. Monzon-Hernandez, V. P. Minkovich, J. Villatoro, M. P. Kreuzer, and G. Badenes, “Photonic crystal fiber microtaper supporting two selective higher-order modes with high sensitivity to gas molecules,” Appl. Phys. Lett. **93**, 081106/1–3 (2008).
[CrossRef]

10. A. Hassani and M. Skorobogatiy, “Design of the microstructured optical fiber-based surface plasmon resonance sensors with enhanced microfluidics,” Opt. Express **14**, 11,616–11,621 (2006).
[CrossRef]

11. B. Gauvreau, A. Hassani, M. F. Fehri, A. Kabashin, and M. A. Skorobogatiy, “Photonic bandgap fiber-based surface plasmon resonance sensors,” Opt. Express **15**, 11,413–11,426 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-18-11413.
[CrossRef]

12. N. M. Litchinitser and E. Poliakov, “Antiresonant guiding microstructured optical fibers for sensing applications,” Appl. Phys. B **81**, 347–351 (2005).
[CrossRef]

13. D. Noordegraaf, L. Scolari, J. Laegsgaard, T. T. Alkeskjold, G. Tartarini, E. Borelli, P. Bassi, J. Li, and S.-T. Wu, “Avoided-crossing-based liquid-crystal photonic-bandgap notch filter,” Opt. Lett. **33**, 986–988 (2008).
[CrossRef] [PubMed]

14. D. K. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. **34**, 322–324 (2009).
[CrossRef] [PubMed]

^{4}nm/refractive index unit (RIU) and detection limits below 10

^{-6}RIU can be achieved [14

14. D. K. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. **34**, 322–324 (2009).
[CrossRef] [PubMed]

16. J. M. Lázaro, B. T. Kuhlmey, J. C. Knight, J. M. Lopez-Higuera, and B. J. Eggleton, “Ultrasensitive UV-tunable grating in all-solid photonic bandgap fibers,” Opt. Commun. **282**, 2358–2361 (2009).
[CrossRef]

17. B. T. Kuhlmey, B. J. Eggleton, and D. K. C. Wu, “Fluid-filled solid-core photonic bandgap fibers,” J. Lightwave Technol. **27**, 1617–1630 (2009).
[CrossRef]

19. B. T. Kuhlmey, F. Luan, J. M. Lazaro, L. Fu, B. J. Eggleton, D. Yeom, S. Coen, A. Wang, J. C. Knight, C. M. B. Cordeiro, and C. J. S. de Matos, “Applications of long period gratings in solid core photonic bandgap fibers,” AIP Conference Proceedings **1055**, 61–64 (2008). URL http://link.aip.org/link/?APC/1055/61/1.
[CrossRef]

20. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express **14**, 10,851–10,864 (2006).
[CrossRef]

10. A. Hassani and M. Skorobogatiy, “Design of the microstructured optical fiber-based surface plasmon resonance sensors with enhanced microfluidics,” Opt. Express **14**, 11,616–11,621 (2006).
[CrossRef]

11. B. Gauvreau, A. Hassani, M. F. Fehri, A. Kabashin, and M. A. Skorobogatiy, “Photonic bandgap fiber-based surface plasmon resonance sensors,” Opt. Express **15**, 11,413–11,426 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-18-11413.
[CrossRef]

20. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express **14**, 10,851–10,864 (2006).
[CrossRef]

4. A. Amezcua-Correa, A. C. Peacock, C. E. Finlayson, J. J. Baumberg, J. Yang, S. M. Howdle, and P. J. A. Sazio, “Surface enhanced Raman scattering using metal modified microstructured optical fibre substrates,” in *32nd European Conference on Optical Communication*, p. Tu4.3.4 (Cannes, France, 2006).
[CrossRef]

21. P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D.-J. Won, F. Zhang, E. R. Margine, V. Gopalan, V. H. Crespi, and J. V. Badding, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science **311**, 1583–1586 (2006).
[CrossRef] [PubMed]

_{2}can be deposited using SOL-GEL techniques by flushing the reactants through the holes [22

22. C. Jing, Xiujian, J. Zhao, K. Han, A. Zhu, H. Liu, and Tao, “A new method of fabricating internally sol-gel coated capillary tubes,” Surf. Coat. Technol. **162**, 228–233 (2003). URL http://www.sciencedirect.com/science/article/B6TVV-472BJXN-6/2/ee5f0e6b9e51b138e7fa5c8fb81d42a4.
[CrossRef]

14. D. K. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. **34**, 322–324 (2009).
[CrossRef] [PubMed]

## 2. Cutoffs and Sensitivity

### 2.1. Sensitivity and detection limit

23. I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express **16**, 1020–1028 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-2-1020.
[CrossRef] [PubMed]

*λ*at a wavelength

*λ*

_{r}depending on the refractive index of the analyte

*n*

_{a}. The sensitivity is defined as

8. L. Rindorf and O. Bang, “Sensitivity of photonic crystal fiber grating sensors: biosensing, refractive index, strain, and temperature sensing,” J. Opt. Soc. Am. B **25**, 310–324 (2008).
[CrossRef]

*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),

*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

*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*.

12. N. M. Litchinitser and E. Poliakov, “Antiresonant guiding microstructured optical fibers for sensing applications,” Appl. Phys. B **81**, 347–351 (2005).
[CrossRef]

13. D. Noordegraaf, L. Scolari, J. Laegsgaard, T. T. Alkeskjold, G. Tartarini, E. Borelli, P. Bassi, J. Li, and S.-T. Wu, “Avoided-crossing-based liquid-crystal photonic-bandgap notch filter,” Opt. Lett. **33**, 986–988 (2008).
[CrossRef] [PubMed]

15. P. Steinvurzel, E. D. Moore, E. C. Mägi, and B. J. Eggleton, “Tuning properties of long period gratings in photonic bandgap fibers,” Opt. Lett. **31**, 2103–2105 (2006).
[CrossRef] [PubMed]

16. J. M. Lázaro, B. T. Kuhlmey, J. C. Knight, J. M. Lopez-Higuera, and B. J. Eggleton, “Ultrasensitive UV-tunable grating in all-solid photonic bandgap fibers,” Opt. Commun. **282**, 2358–2361 (2009).
[CrossRef]

**34**, 322–324 (2009).
[CrossRef] [PubMed]

### 2.2. Solid core photonic bandgap sensors

24. T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. **27**, 1977–1979 (2002).
[CrossRef]

25. J. Laegsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A— Pure Appl. Opt. **6**, 798–804 (2004).
[CrossRef]

_{0q}and LP

_{1q}modes of the individual waveguides formed by the high-index analyte channels [24

24. T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. **27**, 1977–1979 (2002).
[CrossRef]

*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*=2.405, 3.832, … [26]. When the refractive index of the analyte

_{c}*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

15. P. Steinvurzel, E. D. Moore, E. C. Mägi, and B. J. Eggleton, “Tuning properties of long period gratings in photonic bandgap fibers,” Opt. Lett. **31**, 2103–2105 (2006).
[CrossRef] [PubMed]

16. J. M. Lázaro, B. T. Kuhlmey, J. C. Knight, J. M. Lopez-Higuera, and B. J. Eggleton, “Ultrasensitive UV-tunable grating in all-solid photonic bandgap fibers,” Opt. Commun. **282**, 2358–2361 (2009).
[CrossRef]

13. D. Noordegraaf, L. Scolari, J. Laegsgaard, T. T. Alkeskjold, G. Tartarini, E. Borelli, P. Bassi, J. Li, and S.-T. Wu, “Avoided-crossing-based liquid-crystal photonic-bandgap notch filter,” Opt. Lett. **33**, 986–988 (2008).
[CrossRef] [PubMed]

*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.

^{-6}[16

**282**, 2358–2361 (2009).
[CrossRef]

*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

20. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express **14**, 10,851–10,864 (2006).
[CrossRef]

*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

**34**, 322–324 (2009).
[CrossRef] [PubMed]

*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).

## 3. Cutoffs of coated cylinders

27. A. C. Boucouvalas and C. D. Papageorgiou, “Cutoff frequencies in optical fibers of arbitrary refractive-index profile using the resonance technique,” IEEE J. Quantum Elec. **18**, 2027–2031 (1982).
[CrossRef]

29. E. Sharma, I. Goyal, and A. Ghatak, “Calculation of cutoff frequencies in optical fibers for arbitrary profiles using the matrix method,” IEEE J. Quantum Elec. **17**, 2317–2321 (1981).
[CrossRef]

30. A. C. Boucouvalas, “Mode-cutoff frequencies of coaxial optical couplers,” Opt. Lett. **10**, 95–97 (1985).
[CrossRef] [PubMed]

31. E. Karadeniz and P. Kornreich, “Optical fibers with high-index-contrast dielectric thin films,” Opt. Eng. **45**, 105,006 (2006).
[CrossRef]

### 3.1. Fields and modes

*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

*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

*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

*J*(

_{m}*z*) and

*H*

^{(1)}

*(*

_{m}*z*) are the Bessel and Hankel functions of the first kind respectively, and

*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.

*A*are standing waves sourced outside of the coated cylinder, while the terms in

^{c}_{m}*B*

^{a}

*have a singularity at the origin: for a mode all these coefficients are zero [33*

_{m}33. T. White, B. Kuhlmey, R. McPhedran, D. Maystre, G. Renversez, C. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002).
[CrossRef]

*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

*n*

_{eff}=

*β*=

*k*.

_{0}### 3.2. Boundary conditions, scattering matrices, and modal equation

*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

*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}

*=0 and*

_{m}**B**

^{a}

*=0. We thus have*

_{m}^{++}

_{ab})≠0 is equivalent to

### 3.3. Cutoffs

*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:

*m*=0,

*m*=1, and m≥2.

*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.

_{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

*J*and

_{m}*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}),

**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

*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

*V*is the normalised cutoff frequency of the considered mode. Such a function reaches values very close to

_{c}*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

*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

*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*=1.45; second a lower index background material such as Poly(heptafluorobutyl methacrylate) with

_{c}*n*

_{c}=1.383 [18].

### 4.1. Silica background

*V*of the first (TE) and second (TM)

_{c}*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

**34**, 322–324 (2009).
[CrossRef] [PubMed]

**34**, 322–324 (2009).
[CrossRef] [PubMed]

^{-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

8. L. Rindorf and O. Bang, “Sensitivity of photonic crystal fiber grating sensors: biosensing, refractive index, strain, and temperature sensing,” J. Opt. Soc. Am. B **25**, 310–324 (2008).
[CrossRef]

*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

35. G. W. Milton, *The Theory of Composites* (Cambridge University Press, 2002).
[CrossRef]

*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.

*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

**34**, 322–324 (2009).
[CrossRef] [PubMed]

*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

*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.

^{-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

*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.

**14**, 10,851–10,864 (2006).
[CrossRef]

_{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.

*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.

_{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

**34**, 322–324 (2009).
[CrossRef] [PubMed]

^{4}nm/RIU and, assuming similar resonance widths as in [14

**34**, 322–324 (2009).
[CrossRef] [PubMed]

^{-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

**34**, 322–324 (2009).
[CrossRef] [PubMed]

## 6. Conclusion and discussion

^{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.

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

*EK*) component corresponds to the 1st (

*E*) line, 2nd (

*K*) column of the matrix.

## Acknowledgements

## References and links

1. | T. M. Monro, W. Belardi, K. Furusawa, J. C. Baggett, N. G. R. Broderick, and D. J. Richardson, “Sensing with microstructured optical fibres,” Meas. Sci. Technol. |

2. | T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer, and P. J. Sadler, “Quantitative broadband chemical sensing in air-suspended solid-core fibers,” J. Appl. Phys. |

3. | S. Afshar, S. C. Warren-Smith, and T. M. Monro, “Enhancement of fluorescence-based sensing using microstructured optical fibres,” Opt. Express |

4. | A. Amezcua-Correa, A. C. Peacock, C. E. Finlayson, J. J. Baumberg, J. Yang, S. M. Howdle, and P. J. A. Sazio, “Surface enhanced Raman scattering using metal modified microstructured optical fibre substrates,” in |

5. | D. Pristinski and H. Du, “Solid-core photonic crystal fiber as a Raman spectroscopy platform with a silica core as an internal reference,” Opt. Lett. |

6. | L. Rindorf, J. B. Jensen, M. Dufva, L. H. Pedersen, P. E. Hoiby, and O. Bang, “Photonic crystal fiber long-period gratings for biochemical sensing,” Opt. Express |

7. | L. Rindorf and O. Bang, “Highly sensitive refractometer with a photonic-crystal-fiber long-period grating,” Opt. Lett. |

8. | L. Rindorf and O. Bang, “Sensitivity of photonic crystal fiber grating sensors: biosensing, refractive index, strain, and temperature sensing,” J. Opt. Soc. Am. B |

9. | D. Monzon-Hernandez, V. P. Minkovich, J. Villatoro, M. P. Kreuzer, and G. Badenes, “Photonic crystal fiber microtaper supporting two selective higher-order modes with high sensitivity to gas molecules,” Appl. Phys. Lett. |

10. | A. Hassani and M. Skorobogatiy, “Design of the microstructured optical fiber-based surface plasmon resonance sensors with enhanced microfluidics,” Opt. Express |

11. | B. Gauvreau, A. Hassani, M. F. Fehri, A. Kabashin, and M. A. Skorobogatiy, “Photonic bandgap fiber-based surface plasmon resonance sensors,” Opt. Express |

12. | N. M. Litchinitser and E. Poliakov, “Antiresonant guiding microstructured optical fibers for sensing applications,” Appl. Phys. B |

13. | D. Noordegraaf, L. Scolari, J. Laegsgaard, T. T. Alkeskjold, G. Tartarini, E. Borelli, P. Bassi, J. Li, and S.-T. Wu, “Avoided-crossing-based liquid-crystal photonic-bandgap notch filter,” Opt. Lett. |

14. | D. K. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. |

15. | P. Steinvurzel, E. D. Moore, E. C. Mägi, and B. J. Eggleton, “Tuning properties of long period gratings in photonic bandgap fibers,” Opt. Lett. |

16. | J. M. Lázaro, B. T. Kuhlmey, J. C. Knight, J. M. Lopez-Higuera, and B. J. Eggleton, “Ultrasensitive UV-tunable grating in all-solid photonic bandgap fibers,” Opt. Commun. |

17. | B. T. Kuhlmey, B. J. Eggleton, and D. K. C. Wu, “Fluid-filled solid-core photonic bandgap fibers,” J. Lightwave Technol. |

18. | V. Galiatsatos, R. O. Neaffer, S. Sen, and B. J. Sherman, |

19. | B. T. Kuhlmey, F. Luan, J. M. Lazaro, L. Fu, B. J. Eggleton, D. Yeom, S. Coen, A. Wang, J. C. Knight, C. M. B. Cordeiro, and C. J. S. de Matos, “Applications of long period gratings in solid core photonic bandgap fibers,” AIP Conference Proceedings |

20. | B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express |

21. | P. J. A. Sazio, A. Amezcua-Correa, C. E. Finlayson, J. R. Hayes, T. J. Scheidemantel, N. F. Baril, B. R. Jackson, D.-J. Won, F. Zhang, E. R. Margine, V. Gopalan, V. H. Crespi, and J. V. Badding, “Microstructured optical fibers as high-pressure microfluidic reactors,” Science |

22. | C. Jing, Xiujian, J. Zhao, K. Han, A. Zhu, H. Liu, and Tao, “A new method of fabricating internally sol-gel coated capillary tubes,” Surf. Coat. Technol. |

23. | I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express |

24. | T. P. White, R. C. McPhedran, C. M. de Sterke, N. M. Litchinitser, and B. J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. |

25. | J. Laegsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A— Pure Appl. Opt. |

26. | A. Snyder and J. Love, |

27. | A. C. Boucouvalas and C. D. Papageorgiou, “Cutoff frequencies in optical fibers of arbitrary refractive-index profile using the resonance technique,” IEEE J. Quantum Elec. |

28. | W. Gambling, D. Payne, and H. Matsumura, “Cut-off frequency in radially inhomogeneous single-mode fibre,” Elec. Lett. |

29. | E. Sharma, I. Goyal, and A. Ghatak, “Calculation of cutoff frequencies in optical fibers for arbitrary profiles using the matrix method,” IEEE J. Quantum Elec. |

30. | A. C. Boucouvalas, “Mode-cutoff frequencies of coaxial optical couplers,” Opt. Lett. |

31. | E. Karadeniz and P. Kornreich, “Optical fibers with high-index-contrast dielectric thin films,” Opt. Eng. |

32. | M. Abramowitz and I. A. Stegun, |

33. | T. White, B. Kuhlmey, R. McPhedran, D. Maystre, G. Renversez, C. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B |

34. | Wolfram Research, Inc., “Mathematica version 6,” (2007). |

35. | G. W. Milton, |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2370) Fiber optics and optical communications : Fiber optics sensors

(280.1415) Remote sensing and sensors : Biological sensing and sensors

(060.4005) Fiber optics and optical communications : Microstructured fibers

(060.5295) Fiber optics and optical communications : Photonic crystal fibers

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: June 10, 2009

Revised Manuscript: August 18, 2009

Manuscript Accepted: August 23, 2009

Published: August 28, 2009

**Virtual Issues**

Vol. 4, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Boris T. Kuhlmey, Stéphane Coen, and Sahand Mahmoodian, "Coated photonic bandgap fibres for low-index sensing applications: cutoff analysis," Opt. Express **17**, 16306-16321 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-16306

Sort: Year | Journal | Reset

### References

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- T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer, and P. J. Sadler, "Quantitative broadband chemical sensing in air-suspended solid-core fibers," J. Appl. Phys. 103, 103108/1-7 (2008). [CrossRef]
- S. Afshar, S. C. Warren-Smith, and T. M. Monro, "Enhancement of fluorescence-based sensing using microstructured optical fibres," Opt. Express 15, 17,891-17,901 (2007).
- A. Amezcua-Correa, A. C. Peacock, C. E. Finlayson, J. J. Baumberg, J. Yang, S. M. Howdle, and P. J. A. Sazio, "Surface enhanced Raman scattering using metal modified microstructured optical fibre substrates," in 32nd European Conference on Optical Communication, p. Tu4.3.4 (Cannes, France, 2006). [CrossRef]
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- L. Rindorf, J. B. Jensen, M. Dufva, L. H. Pedersen, P. E. Hoiby, and O. Bang, "Photonic crystal fiber long-period gratings for biochemical sensing," Opt. Express 14, 8224-8231 (2006). [CrossRef] [PubMed]
- L. Rindorf and O. Bang, "Highly sensitive refractometer with a photonic-crystal-fiber long-period grating," Opt. Lett. 33, 563-565 (2008). [CrossRef] [PubMed]
- L. Rindorf and O. Bang, "Sensitivity of photonic crystal fiber grating sensors: biosensing, refractive index, strain, and temperature sensing," J. Opt. Soc. Am. B 25, 310-324 (2008). [CrossRef]
- D. Monzon-Hernandez, V. P. Minkovich, J. Villatoro, M. P. Kreuzer, and G. Badenes, "Photonic crystal fiber microtaper supporting two selective higher-order modes with high sensitivity to gas molecules," Appl. Phys. Lett. 93, 081106/1-3 (2008). [CrossRef]
- A. Hassani and M. Skorobogatiy, "Design of the microstructured optical fiber-based surface plasmon resonance sensors with enhanced microfluidics," Opt. Express 14, 11,616-11,621 (2006). [CrossRef]
- B. Gauvreau, A. Hassani, M. F. Fehri, A. Kabashin, and M. A. Skorobogatiy, "Photonic bandgap fiber-based surface plasmon resonance sensors," Opt. Express 15, 11,413-11,426 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-18-11413. [CrossRef]
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- D. Noordegraaf, L. Scolari, J. Laegsgaard, T. T. Alkeskjold, G. Tartarini, E. Borelli, P. Bassi, J. Li, and S.-T. Wu, "Avoided-crossing-based liquid-crystal photonic-bandgap notch filter," Opt. Lett. 33, 986-988 (2008). [CrossRef] [PubMed]
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