We propose and theoretically demonstrate a highly sensitive optofluidic refractive index (RI) sensor based on a spectral filter formed by a segment of liquid-filled seven-hole Teflon-cladding fiber sandwiched by two standard single mode fibers (SMFs). When liquid flows through the air hole channels of the seven-hole Teflon-cladding fiber, it forms a seven-liquid-core fiber (SLCF) and the lightwaves are well guided by the liquid cores owing to total inner reflection. When the input SMF is aligned to the central core of the SLCF, the light excited in the central core will couple to outer cores periodically along the length of the SCLF. At the detection port, the output SMF is also aligned to the central core of the SLCF. Since the coupling coefficient depends on wavelength, the coupling efficiency is also wavelength dependent, leading to a filter spectrum for a given length of the SLCF. The spectral response of the filter to the change in RI of the liquid cores is numerically simulated based on the coupled-mode theory through finite-element method. The dependence of the RI sensitivity on the diameter and pitch of air holes of the SLCF are studied, respectively. Finally, a very high sensitivity of 25,300 nm/RIU for RI around 1.333 is achieved.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Fiber-optic refractive index (RI) sensors have been extensively studied because they are very useful for label-free biological and biochemical sensing [1,2] due to their advantages of high sensitivity, small size, good bio-compatibility, remote sensing, fast response, and immunity to electromagnetic interferences. Most of them are based on the evanescent field effect of optical fibers as well as the components therein. For instance, long-period fiber gratings , optical microfiber devices , and single-multi-single mode (SMS) fiber modal interferometers  were demonstrated for RI sensing.
In recent years, optofluidic RI sensors have attracted much interest because they combine the advantages of optical sensors and microfluidics, i.e., high sensitivity, label-free sensing, fast response, and very few sample consumption . Microstructured optical fibers (MOFs) in which micron-sized air holes are running along their length provide a promising platform for the development of novel fiber-optic microfluidic RI sensors [7–9]. The fluids and lightwaves in MOFs can overlap strongly and interact with each other over a long length. The liquid sample consumption can be as less as several nano-liters per centimeter of MOF.
Various MOF-based optofluidic sensors have been reported [10–20], such as those based on Bragg gratings, long-period gratings, directional coupling, modal interference, and Fabry-Perot interferometers. In 2008, Lindorf and Bang reported a photonic crystal fiber long-period grating RI sensor with a sensitivity of 1500 nm/RIU for RI around 1.33 . In 2009, Wu et al. reported an ultrasensitive photonic crystal fiber based on directional coupling of the core mode to a neighbor selectively-filled liquid core with a sensitivity of 30,200 nm/RIU . However, it can only sense the liquid with RI higher than that of the background material, i.e., 1.444 for pure silica. In 2011, Sun et al. proposed a microfluidic RI sensor with a sensitivity of 8500 nm/RIU for RI around 1.33 based on resonant coupling from the silica core to the microstructure-core of a specially designed MOF, in which the microstructure-core was formed by selectively filling several specific air holes of the MOF with liquid . In 2013, our group reported a microfluidic RI sensor with a sensitivity of 5600 nm/RIU for RI around 1.33 based on a high-birefringence MOF Sagnac interferometer . In 2016, Zhang et al. reported a microfluidic RI sensor with a sensitivity of 1145 nm/RIU for RI from 1.33 to 1.39 based on modal interference of the LP01 and LP11 modes of a side-channel MOF . In 2018, Yang et al. demonstrated an in-fiber Mach-Zehnder interferometer with a sensitivity of 354 nm/RIU using a hole-assisted two-core fiber . Most of reported microfluidic RI sensors either exhibit relatively low sensitivity or require selectively liquid filling technique.
Recently, multicore fibers, especially seven-core fibers (SCFs), have been attracting much interest for a variety of sensing applications of high temperature, curvature, strain, torsion, etc. [21–26]. Owing to the wavelength dependent coupling from the central core to the outer cores of multicore fibers, it is easy to construct high quality spectral filters by inserting a piece of multicore fiber between two standard single-mode fibers (SMFs) via central alignment at both ends. Particularly, the SCF filters have polarization-independent transmission spectrum , which is unlike two-core fiber filters . On the other hand, Teflon as cladding material has been widely used for liquid-core waveguides and fibers because its refractive index is slightly lower than that of water and it has good bio-compatibility [28–30]. The combination of the SCF structure and the Teflon material is a promising solution for the development of new optofluidic RI sensors since it can inherit the advantages of SCF-based sensors.
In this paper, we propose the design of a novel seven-hole Teflon-cladding fiber. A seven-liquid-core fiber (SLCF) can be formed by filling all the holes of this fiber with flowing liquid. Then a spectral filter can be achieved by inserting a segment of SLCF between two standard SMFs. The change in the RI of the liquid cores causes the variation of coupling coefficient between the central core and outer cores of the SLCF, leading to wavelength shift of the transmission spectrum of the filter, and hence an optofluidic RI sensor can be obtained. The response of the filter’s transmission spectrum to the change in RI of the liquid cores is numerically simulated based on the coupled-mode theory through finite-element method. The dependence of the RI sensitivity on the diameter and pitch of air holes of the SLCF and the operation wavelength are studied, respectively. Finally, a very high sensitivity of 25,300 nm/RIU for liquid RI around 1.333 is achieved. The proposed sensor is attractive for label-free biological and biochemical sensing owing to its high sensitivity and very few sample consumption.
2. Principle of operation
The cross section geometry of the proposed seven-hole Teflon-cladding fiber is shown in Fig. 1(a). Its outer diameter is 125 µm, the same as that of a standard SMF. It has seven air holes, with one in the center surrounded by the other six hexagonally arranged air holes. All the air holes have an identical diameter d. The center-to-center distances of all adjacent air holes are equal and defined as the air hole pitch Λ. The cladding material of the fiber is Teflon AF-1300, whose refractive index is slightly lower than that of water. The refractive indices of Teflon AF-1300 and pure water as a function of wavelength in the wavelength range from 1400 nm to 1650 nm are plotted in Fig. 1(b), which are calculated by Sellmeier equations [30,31]. The proposed seven-hole Teflon-cladding fiber can be drawn from a seven-hole Teflon preform. The preform can be fabricated by axially drilling seven holes in a Teflon rod.
A piece of SLCF is inserted between two standard SMFs with the assistance of a double-T shaped microfluidic chip as Fig. 1(c) shows. The height and width of the microfluidic channel is 125 µm by 125 µm so that the SMFs and the SLCF are centrally aligned. There is a gap of around 10 µm between the fiber ends so that the fluid can run into and out of the air holes of the SLCF. When water or aqueous solution flows through all the air hole channels of the fiber, it forms a seven-liquid-core fiber (SLCF). Light waves can be well guided in the liquid cores due to total inner reflection.
The coupling characteristics of multicore fibers with one central core and l circularly distributed outer cores (i.e., the 1 + l core structure) under various launching conditions were systematically and comprehensively studied by Kishi and Yamashita based on the coupled-mode theory . In their theory, the coupling coefficients between nonadjacent cores were neglected because they were considered very small in the first order approximation. Coupling coefficients between two adjacent cores were obtained from a dual-core structure whose core separation equals that of adjacent two cores of a multicore structure. The core diameter and refractive index profile of the dual-core structure were the same as that of the individual core of the multicore fiber. If the separations between the outer two adjacent cores and between the central core and outer cores of a multicore fiber are different, one has to study two dual-core structures. For the proposed SLCF, it is a 1 + 6 core structure, and the separations between all adjacent cores of it are the same. So we only have to study a single dual-core structure.
In order to illustrate the principle of operation of the proposed sensor, we firstly study an SLCF with d = 6 µm, Λ = 8 µm as an example. We assume the initial liquid is pure water with nominal RI nliquid = 1.333 (the nominal RI refers to its value at the wavelength of 589.3 nm, since material RI depends on wavelength). Using the perpendicular wave module of COMSOL Multiphysics (a commercially available finite-element method software), we simulate the supermodes of a dual-core fiber with d = 6 µm, Λ = 8 µm, ncore = nliquid, and nclad = nTeflon. We can achieve the mode fields and effective indices as well as their wavelength dependence of all the supermodes of it, as shown in Fig. 2. The material dispersions of water and Teflon AF-1300 are considered based on the Sellmeier equations of them [30,31].
From Fig. 2 we obtain that the effective index difference for x- and y-polarized odd and even supermodes of the dual-core fiber are calculated to be Δneff,xoe = 5.24 × 10−4 and Δneff,yoe = 5.21 × 10−4, respectively. The difference of the above two values is about 0.6%, indicating very weak polarization dependence of the fiber, so we only consider x-polarized supermodes in later simulations. The coupling coefficient of a dual-core fiber can be obtained by calculating the difference in the propagation constants of its odd and even supermodes as follow [32,33]:
Since the input SMF and the SLCF are center to center aligned, the launching condition is assumed to be I0,0 = 1 for the 0th central core, and Ii,0 = 0 for the ith outer core (i = 1, 2, 3, 4, 5, 6). Then we obtain that, the electrical field intensity evolution along the length of the SLCF for each individual core is given by the following equations :
For an SLCF filter with length of L, the optical power in the central core collected by the output SMF is given by the following equation:
We notice that, Eq. (3) is consistent with the formulation presented in , which is also deduced from the coupled-mode theory . For a certain transmission dip with wavelength λm of the spectrum described by Eq. (3), its phase term keeps to be a constant of (2m+1)π/2 with the change of nliquid, where m is an integer. For RI sensing, L is assumed to be a constant. Then we obtain the following equation:
Differentiating both sides of Eq. (2) with respect to nliquid, we can obtain that the RI sensitivity of the filter is determined by three factors as shown in the following equation:
3. Results and discussion
3.1 Transmission spectrum
Equation (3) indicates that the transmission spectrum of the filter is mainly determined by Δneffoe(λ) of the odd and even supermodes of the dual-core fiber with the same core diameter and core separation. So we change the operation wavelength with a step of 25 nm over the wavelength range from 1450 nm to 1650 nm, and repeat the simulation using COMSOL software. By doing this we can obtain Δneffoe as a function of λ for the dual-core fiber with d = 6 µm, Λ = 8 µm, and nliquid = 1.333 as shown in Fig. 3(a). Then we perform six-order polynomial fitting for Δneffoe and λ, achieving the numerical relationship of them with R2 = 0.99999998. Finally, we can simulate the transmission spectrum of the filter by substituting the achieved Δneffoe(λ) equation into Eq. (3), setting L = 5 mm, and performing logarithmic calculation, as shown in Fig. 3(b). The dissimilarity of the numerical apertures of the SLCF and the standard SMF will leads to additional insert loss of the device. Since the proposed sensor is wavelength encoded, the insert loss and material absorption loss of the device are not considered. The maximum output power is normalized to be 1 (0 dB). Here, the wavelength resolution of the spectrum is set to be 0.02 nm, the same as that of a common commercial optical spectrum analyzer.
3.2 Optofluidic RI response
In order to study the RI response of the sensor, we change the liquid refractive index with a step of 0.0005, and repeat the wavelength scanning simulation as done in section 3.1. Note that we assume the liquid refractive index change is the same over the whole wavelength range. Then we obtain Δneffoe as a function of wavelength for different values of nliquid as shown in Fig. 4(a). After that, we can obtain the transmission spectra for different values of nliquid as shown in Fig. 4(b). By tracking the wavelength shifts of the two transmission notches of Fig. 4(b), we obtain spectral notch wavelength as a function of nliquid, as shown in Fig. 4(c). The spectrum shifts towards longer wavelengths with the increasing of nliquid. Then we do linear fitting of the response curves and obtain very high RI sensitivities of 20,332 nm/RIU with R2 = 0.99993 and 20,500 nm/RIU with R2 = 0.99995 for the wavelength bands of ∼1490 nm and ∼1570 nm, respectively. This indicates that the RI sensitivity exhibits very weak wavelength dependence.
3.3 Dependence of RI sensitivity on the air hole diameter d
Equation (3) shows that the RI sensitivity is mainly determined by the properties of Δneffoe of the dual-core fiber, which depends much on the geometrical parameters of the fiber. Here, we fix the air hole pitch to be Λ = 10 µm, and change the air hole diameter d from 3 µm to 9 µm with a step of 1 µm, i.e., d/Λ varies from 0.3 to 0.9 with a step of 0.1.
Through simulation and calculation, we find that, when d increases, Δneffoe first decreases and then increases as shown in Fig. 5(a). With the decreasing of d, the evanescent fields of individual cores become stronger (leading to stronger coupling between adjacent cores and giving rising to larger Δneffoe), whereas the edge to edge distance between adjacent cores becomes larger (leading to weaker coupling between adjacent cores and giving rise to smaller Δneffoe). There is a trade-off between these two effects and d = 5 µm is the inflection point for SLCF with Λ = 10 µm. When d is smaller than 5 µm, the evanescent field intensity dominates the evolution of Δneffoe; when d is larger than 5 µm, the edge to edge distance between adjacent cores dominates the evolution of Δneffoe.
There are also inflection points for the evolutions of ∂Δneffoe/∂nliquid and Goe with the increasing of d, as shown in Fig. 5(b) and (c), leading to non-monotonic evolution of the RI sensitivity with the increasing of d. Since both the values of ∂Δneffoe/∂nliquid and Goe are negative, according to Eq. (5), the RI sensitivity exhibits positive values as shown in Fig. 5(d). We find that, when d = 4 µm, a highest RI sensitivity of 24,227 nm/RIU for SLCF with Λ = 10 µm is achieved. A moderate d/Λ helps to achieve higher RI sensitivity. A too small or too large d/Λ leads to degradation of the RI sensitivity.
3.4 Dependence of RI sensitivity on the air hole pitch Λ
In order to study how the air hole pitch of SLCF affects the RI sensitivity, we change Λ to be 8 µm and 12 µm, and repeat the simulation process as what we do in section 3.3, respectively. The results are shown in Fig. 6. We find that, over the whole d/Λ range from 0.3 to 0.9, the RI sensitivity increases with the increasing of Λ. A highest RI sensitivity of 25,300 nm/RIU is achieved when Λ = 12 µm and d = 4.8 µm. For an SLCF with a smaller Λ, its RI sensitivity is more sensitive to the change of d/Λ.
With the increasing of Λ, the coupling between adjacent cores becomes weaker, leading to the decreasing of Δneffoe and the absolute value of Goe. Equation (3) indicates that the RI sensitivity is inversely proportional to the absolute value of Goe. Hence, a larger Λ gives rise to higher RI sensitivity. However, if we further increase Λ, the coupling between adjacent cores will become too weak. For example, when Λ = 16 µm and d = 6.4 µm, Δneffoe decreases to 1.3 × 10−6 which is two orders of magnitude lower than that of the SLCF with Λ = 8 µm and d = 3.2 µm.
3.5 Dependence of RI sensitivity on the operation wavelength λ
In previous two sections, the simulations are performed at the wavelength of 1550 nm. Here, we set d/Λ = 0.4, Λ = 8 µm, 10 µm, and 12 µm, respectively. For each case, we do wavelength scanning and achieve the wavelength dependence of RI sensitivity of them, as shown in Fig. 7. The results indicates that, for all these three fiber structures, the fluctuation of RI sensitivity is less than 5% over a broad wavelength range from 1450 nm to 1650 nm.
A highly sensitive optofluidic RI sensor based on a novel seven-hole Teflon-cladding fiber is proposed. The integration of the seven-hole Teflon-cladding fiber with liquid fluids forms an SLCF, and a spectral filter is constructed by inserting it between two SMFs. The coupling characteristics and optofluidic RI response of the filter are theoretically studied based on the coupled-mode theory through finite element method. The proposed sensor has high sensitivity and good linearity. The dependence of its RI sensitivity on the structural parameters of SLCF are studied and discussed. Finally, we find that, for an SLCF with Λ = 12 µm and d = 4.8 µm, a very high sensitivity of 25,300 nm/RIU for RI around 1.333 is achieved.
Natural Science Foundation of Guangdong Province (2014A030306040, 2018A030313440); Tip-top Scientific and Technical Innovative Youth Talents of Guangdong Special Support Program (2016TQ03X124, 2019TQ05X136); Pearl River S and T Nova Program of Guangzhou (201806010197); The Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program (2019BT02X105); Guangzhou Science and Technology Plan Project (201904020032).
The authors declare no conflicts of interest.
1. X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Anal. Chim. Acta 620(1-2), 8–26 (2008). [CrossRef]
2. X. Wang and O. Wolfbeis, “Fiber-optic chemical sensors and biosensors (2013–2015),” Anal. Chem. 88(1), 203–227 (2016). [CrossRef]
3. S. W. James and R. P. Tatam, “Optical fibre long-period grating sensors: characteristics and application,” Meas. Sci. Technol. 14(5), R49–R61 (2003). [CrossRef]
4. J. H. Chen, D. R. Li, and F. Xu, “Optical microfiber sensors: sensing mechanisms, and recent advances,” J. Lightwave Technol. 37(11), 2577–2589 (2019). [CrossRef]
5. Q. Wu, Y. Semenova, P. Wang, and G. Farrell, “High sensitivity SMS fiber structure based refractometer – analysis and experiment,” Opt. Express 19(9), 7937–7944 (2011). [CrossRef]
6. X. Fan and I. M. White, “Optofluidic microsystems for chemical and biological analysis,” Nat. Photonics 5(10), 591–597 (2011). [CrossRef]
7. B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9(13), 698–713 (2001). [CrossRef]
8. J. M. Fini, “Microstructure fibers for optical sensing in gases and liquids,” Meas. Sci. Technol. 15(6), 1120–1128 (2004). [CrossRef]
9. M. Calcerrada, C. Garcia-Ruiz, and M. González-Herráez, “Chemical and biochemical sensing applications of microstructured optical fiber-based systems,” Laser Photonics Rev. 9(6), 604–627 (2015). [CrossRef]
10. J. B. Jensen, L. H. Pedersen, P. E. Hoiby, L. B. Nielsen, T. P. Hansen, and A. Bjarklev, “Photonic crystal fiber based evanescent-wave sensor for detection of biomolecules in aqueous solutions,” Opt. Lett. 29(17), 1974–1976 (2004). [CrossRef]
11. M. C. Huy, G. Laffont, V. Dewynter, P. Ferdinand, P. Roy, J. L. Auguste, D. Pagnoux, W. Blanc, and B. Dussardier, “Three-hole microstructured optical fiber for efficient fiber Bragg grating refractometer,” Opt. Lett. 32(16), 2390–2392 (2007). [CrossRef]
12. L. Rindorf and O. Bang, “Highly sensitive refractometer with a photonic-crystal-fiber long-period grating,” Opt. Lett. 33(6), 563–565 (2008). [CrossRef]
13. D. K. C. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 34(3), 322–324 (2009). [CrossRef]
14. G. E. Town, W. Yuan, R. McCosker, and O. Bang, “Microstructured optical fiber refractive index sensor,” Opt. Lett. 35(6), 856–858 (2010). [CrossRef]
15. B. Sun, M. Y. Chen, Y. K. Zhang, J. C. Yang, J. Q. Yao, and H. X. Cui, “Microstructured-core photonic-crystal fiber for ultra-sensitive refractive index sensing,” Opt. Express 19(5), 4091–4100 (2011). [CrossRef]
16. J. Tian, Y. Lu, Q. Zhang, and M. Han, “Microfluidic refractive index sensor based on an all-silica in-line Fabry–Perot interferometer fabricated with microstructured fibers,” Opt. Express 21(5), 6633–6639 (2013). [CrossRef]
17. S. H. Kassani, J. Park, Y. Jung, J. Kobelke, and K. Oh, “Fast response in-line gas sensor using C-type fiber and Ge-doped ring defect photonic crystal fiber,” Opt. Express 21(12), 14074–14083 (2013). [CrossRef]
18. C. Wu, M. L. V. Tse, Z. Y. Liu, B. O. Guan, C. Lu, and H. Y. Tam, “In-line microfluidic refractometer based on C-shaped fiber assisted photonic crystal fiber Sagnac interferometer,” Opt. Lett. 38(17), 3283–3286 (2013). [CrossRef]
19. N. Zhang, G. Humbert, Z. Wu, K. Li, P. Shum, N. M. Zhang, Y. Cui, J. L. Auguste, X. Q. Dinh, and L. Wei, “In-line optofluidic refractive index sensing in a side-channel photonic crystal fiber,” Opt. Express 24(24), 27674–27682 (2016). [CrossRef]
20. J. Yang, M. Yang, C. Y. Guan, J. H. Shi, Z. Zhu, P. Li, P. F. Wang, J. Yang, and L. B. Yuan, “In-fiber Mach-Zehnder interferometer with piecewise interference spectrum based on hole-assisted dual-core fiber for refractive index sensing,” Opt. Express 26(15), 19091–19099 (2018). [CrossRef]
21. J. E. Antonio-Lopez, Z. S. Eznaveh, P. LiKamWa, A. Schülzgen, and R. Amezcua-Correa, “Multicore fiber sensor for high-temperature applications up to 1000(C,” Opt. Lett. 39(15), 4309–4312 (2014). [CrossRef]
22. G. Salceda-Delgado, A. Van Newkirk, J. E. Antonio-Lopez, A. Martinez-Rios, A. Schülzgen, and R. Amezcua Correa, “Compact fiber-optic curvature sensor based on super-mode interference in a seven-core fiber,” Opt. Lett. 40(7), 1468–1471 (2015). [CrossRef]
23. M. S. Yoon, S. B. Lee, and Y. G. Han, “In-line interferometer based on intermodal coupling of a multicore fiber,” Opt. Express 23(14), 18316–18322 (2015). [CrossRef]
24. J. Villatoro, O. Arrizabalaga, G. Durana, I. Sáez de Ocáriz, E. Antonio-Lopez, J. Zubia, A. Schülzgen, and R. Amezcua-Correa, “Accurate strain sensing based on super-mode interference in strongly coupled multi-core optical fibres,” Sci. Rep. 7(1), 4451 (2017). [CrossRef]
25. H. Zhang, Z. Wu, P. P. Shum, X. Shao, R. Wang, X. Q. Dinh, S. Fu, W. Tong, and M. Tang, “Directional torsion and temperature discrimination based on a multicore fiber with a helical structure,” Opt. Express 26(1), 544–551 (2018). [CrossRef]
26. F. Tan, Z. Liu, J. Tu, C. Yu, C. Lu, and H. Y. Tam, “Torsion sensor based on inter-core mode coupling in seven-core fiber,” Opt. Express 26(16), 19835–19844 (2018). [CrossRef]
27. J. R. Guzman-Sepulveda and D. A. May-Arrioja, “In-fiber directional coupler for high-sensitivity curvature measurement,” Opt. Express 21(10), 11853–11861 (2013). [CrossRef]
28. R. Altkorn, I. Koev, R. P. V. Duyne, and M. Litorja, “Low-loss liquid-core optical fiber for low-refractive-index liquids: fabrication, characterization, and application in Raman spectroscopy,” Appl. Opt. 36(34), 8992–8998 (1997). [CrossRef]
29. S. H. Cho, J. Godin, and Y. Lo, “Optofluidic waveguides in Teflon AF-coated PDMS microfluidic channels,” IEEE Photonics Technol. Lett. 21(15), 1057–1059 (2009). [CrossRef]
30. M. Yang, R. French, and E. Tokarsky, “Optical properties of Teflon® AF amorphous fluoropolymers,” J. Micro/Nanolith. MEMS MOEMS 7(3), 033010 (2008). [CrossRef]
31. A. H. Harvey, J. S. Gallagher, and J. M. H. L. Sengers, “Revised formulation for the refractive index of water and steam as a function of wavelength, temperature and density,” J. Phys. Chem. Ref. Data 27(4), 761–774 (1998). [CrossRef]
32. N. Kishi and E. Yamashita, “A simple coupled-mode analysis method for multiple-core optical fiber and coupled dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 36(12), 1861–1868 (1988). [CrossRef]
33. N. Mothe and P. D. Bin, “Numerical analysis of directional coupling in dual-core microstructured optical fibers,” Opt. Express 17(18), 15778–15789 (2009). [CrossRef]
34. F. Y. M. Chan, A. P. T. Lau, and H. Y. Tam, “Mode coupling dynamics and communication strategies for multi-core fiber systems,” Opt. Express 20(4), 4548–4563 (2012). [CrossRef]
35. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62(11), 1267–1277 (1972). [CrossRef]