1. Introduction

Optical fibers, which are essentially dielectric structures transmitting electromagnetic waves at optical frequencies, exist in a large variety of forms. They can be natural such as the glass sponge [1] as well as artificially made [2]. Their applications range from long-haul telecommunications to medicine, spectroscopy and sensors. Recently, a new class of optical fibers, microstructured optical fibers (MOFs) or photonic crystal fibers, has been introduced [3–12]. These fibers are made of glass or polymer with a cross-section containing a structure on the scale of a micron. The presence of this structure (often in the form of circular air-holes or concentric cylinders) changes the transmission characteristics of the fiber. Many properties of MOFs are determined by the air-hole size and the spacing between them. Moreover, once the fiber has been made, its properties can be further modified by introducing various materials into the air-holes [13,14]. Using materials with variable thermal or electro-optic properties allows the fibers to be reversibly tuned. Additional degrees of design freedom provided by MOFs are being utilized in various photonic devices including microfluidic fiber gratings [9] or dynamically tunable MOFs [10].

MOFs with a solid core surrounded by air-holes guide light through a modified total internal reflection (TIR) mechanism. Filling the air-holes with high-index material rules out TIR-based guidance, since the refractive index of the core is lower than that of the cladding. Recently, the guiding properties of MOFs consisting of a low refractive index core and air-holes filled with high refractive index material [15–18] were investigated. A spectral response of these fibers is approximately periodic in frequency. A new guiding regime was identified in which the positions of spectral minima are largely determined by the individual properties of high index inclusions rather than their position and number. The physical mechanism of guiding in this regime is similar to that in antiresonant reflecting optical waveguides (ARROW) [19], widely used in the field of integrated optics. A simple analytical model was proposed to predict the locations of spectral minima based on the geometry and refractive indexes of the inclusions.

In a present paper we show how this simple model can be used for manipulating the properties of MOF with high-index inclusions and designing tunable fiber devices. A large variety of tunable photonic devices has been described in the literature including silica fiber-Bragg-grating-based (FBG) filters [20], polymer Bragg gratings [21], tunable filters based on FBG and long-period gratings (LPGs) written in an index-guiding MOF [9,10]. A majority of the filtering applications requires introducing some kind of a resonant structure inside the core of the fiber such as FBGs or LPGs. Once such a resonant structure is introduced, its properties can be tuned only within a certain wavelength range that is determined by the properties of the fiber material (silica or polymer). ARROW-like MOFs with high index inclusions have several advantages for designing tunable devices. First, their spectral response consists of many narrow resonant features that are approximately periodically spaced in frequency. Therefore, no additional resonant structures are required to be introduced. The positions of these resonances are largely determined by the refractive indexes of the inclusions and the background material (silica or polymer), and by the size of the high-index inclusion. Second, ARROW-like MOFs offer a possibility of a more flexible tuning range compared to grating-based devices, which could be achieved by replacing the high-index material in the air-holes. A simple analytical model reviewed here takes into account advanced features of ARROW-like MOFs and thus could be utilized in designing of tunable photonic devices.

 

Fig. 1. (a) MOF with low-index core and high-index inclusions, (b) Corresponding cross-section of the refractive index profile along x axis, (c) Planar optical waveguide with low-index core and high-index layers.

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In the following sections we review a theory that allows prediction of the transmission spectrum minima in MOF with high index inclusions (Section II) and discuss the possibility of designing novel photonic devices utilizing MOF with tunable refractive index of the cylindrical inclusions (Section III). Our findings are summarized in Section IV.

2. Theory

2.1 Planar waveguide: Antiresonant reflecting optical waveguide mode

The cross-section of a typical MOF consisting of a low-index solid core surrounded by ten hexagonal rings of circular air-holes filled with high-index liquid is shown in Fig. 1(a). The refractive index profile of this fiber along x direction is given in Fig. 1(b). Initially we simplify MOF by reducing it to a one-dimensional planar waveguide having a similar refractive index profile as shown in Fig. 1(c) [16]. The propagation of a Gaussian beam in z direction has been investigated using the standard beam propagation method (BPM)[22]. The transmission spectrum is defined as the ratio of the integrated power within the core region at the end of the waveguide to that launched into the core. The main results of the simulations can be summarized as follows. First, the locations of the spectral minima are essentially unaffected by changes in lattice constant, provided that the refractive indexes (n₁, n₂) and the thickness of high-index layer d were kept constant. Second, the locations of the spectral minima remain unchanged when all layers except one on each side of the core were removed. Finally, the transmission minima shifted in either of the following cases: (i) d was changed while n₁ and n₂ were kept constant, or (ii) d was kept constant, while n1 or n2 were changed. These simulations identified a guiding regime in which the properties of high-index layer largely determine the spectral properties of the entire waveguide. In this regime the addition of new layers and changing their spacing results in a better confinement and an appearance of some fine structure near the transmission minima (that becomes increasingly more noticeable at longer wavelengths, see Ref. [16]). These conclusions allow us to simplify the problem by replacing a 10-layer structure in Fig. 1(b) by 1-layer structure shown in Fig. 2(a). Figure 2(b) shows the calculated spectrum for n₁=1.4, n₂=1.8, d=3.437µm and a waveguide length of 5cm. Electric field profiles at wavelengths corresponding to (1) a high transmission at λ=0.676µm, and (2) a transmission minimum at λ=0.707µm are shown in Fig. 2(c). The electric field oscillations inside the high-index layer are enlarged in Fig. 2(d). It is noteworthy that at wavelengths corresponding to transmission minima, the electric field forms a standing wave pattern with an integer number of half-oscillations inside the high-index layer. This suggests that the guiding mechanism in this type of waveguides is very similar to an antiresonant reflecting optical waveguide (ARROW) principle [19]. In ARROW light is confined in the core by antiresonant reflection from the high-index layers in the cladding. Each high-index layer can be considered as a Fabry-Perot-like (FP) resonator. Narrow-band resonances of this FP resonator correspond to transmission minima for the light propagating in the core. Therefore, at resonance the high-index layer becomes effectively transparent and light escapes completely from the entire structure. Wide antiresonances of the FP correspond to high transmission regions.

The resonant condition for a high-index layer is given by k_td=πm, where $k_t=\frac{2\pi }{\lambda }\sqrt{n_2^2-n_1^2}$. We assume that light impinges the core/cladding interface at glancing angles (i.e. λ/a<<1). The wavelengths corresponding to transmission minima are given by:

Obviously, the simple model described here is valid only in a “short wavelength” regime defined by $\lambda ⁄d\le 2\sqrt{n_2^2-n_1^2}$.

 

Fig. 2. (a) A schematic of a waveguide, (b) Corresponding transmission spectrum at a distance of 5cm in z direction, (c) Electric field profile at a distance of 5cm at the wavelength corresponding to high transmission (1) and a transmission minimum (2), (d) Electric field oscillations inside the high-index layer.

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2.2 Microstructured fiber: Scattering resonances

In this section we study the possibility of extending the ARROW model to understand the mechanism for light propagation in MOFs with high-index inclusions embedded in a low-index background, as shown in Fig. 1. In Ref. [17] using a rigorous mutipole method [23,24] we numerically calculated the complex propagation constant β (and the corresponding effective refractive index) of the fundamental mode for various MOF configurations including a three-ring hexagonal structure, a single hexagonal ring structure, a ten-cylinder-ring structure and even ten cylinders arranged randomly around the core. The modal effective refractive index n_eff in these structures always contains a real and an imaginary part since the structures are leaky. The imaginary part of n_eff is proportional to the confinement loss. The results of these simulation can be summarized as follows: although the actual values of the effective refractive index and loss vary for each of the above structures, the shapes of the loss curves are similar and the high-loss regions occur near the same wavelengths for all fiber configurations. The only parameters that were kept constant in all cases were the cylinder diameter d=3.315µm, low refractive index n₁=1.44 (background) and high refractive index n₂=1.8 (inside the cylinders). This suggested that the properties of individual high-index cylinders largely determine the locations of high-loss regions.

This notion was confirmed by studying the scattering of a plane wave at oblique incidence upon the surface of an infinite cylinder (shown in Fig. 3(a)) [25,26]. The scattering properties can be characterized by a scattering cross section and a forward-to-backward scattering ratio which is a measure of the ratio of the energy flowing out of the cylinder to the energy reflected back. Figure 3(b) shows good qualitative agreement between MOF confinement loss and forward-to-backward scattering ratio for a plane wave scattered on a single high-index cylinder. The angle of incidence of the plane wave was varied according to the effective index of the MOF mode at each wavelength.

 

Fig. 3. Comparison of MOF loss properties with forward-to-backward scattering ratio for the plane wave scattering on a single cylinder.

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These results indicate that the modal loss properties of MOFs with high-index inclusions surrounding a low-index cladding can be understood in terms of a plane wave scattering from a single cylinder and are therefore determined by the properties of individual cylinders rather than by their positions and number. Next section describes how to predict the locations of the high-loss regions analytically, in a manner similar to the planar waveguide case.

2.3 Modal cutoff

In previous sections a new guidance regime was described in which the positions of loss minima depend on the properties of individual high-index inclusions. Now we consider each high-index inclusion as a step-index waveguide with a high-index core (with the refractive index n₂) and a low-index cladding (with the refractive index n₁). Let’s start with a planar waveguide shown in Fig. 1(c) first. The properties of a step-index waveguide are well understood. At a given wavelength the waveguide supports N guided modes with modal effective refractive indices in the range n₁<n_eff<n₂. The effective refractive index of the highest order mode supported is the closest to the refractive index of the cladding. As the wavelength increases, n _eff approaches the refractive index of the cladding n₁. A wavelength at which the waveguide switches from supporting n+1 modes to n modes is referred to as modal cutoff wavelength for a particular mode. In a planar waveguide no power of the mode can propagate in the high-index core at cutoff and all the modes’ power is in the cladding. Modal group velocity given by [27]

where c is the speed of light, n₂ is the refractive index of the core, Δ=($n_2^2$-$n_1^2$)/2$n_2^2$, η is the fraction of modal power residing in the core. At cutoff η→0 and, therefore, the modal group velocity approaches the speed of light in the cladding c/n₁.

Next, we establish a link between the properties of a single high-index inclusion at cutoff and the properties of the entire ARROW structure consisting of a low-index core n1 and high-index n2 layers in the cladding (as shown in Fig. 1(c)). The fundamental and other lower order modes of this waveguide impinge the core-cladding interface at glancing angles (assuming λ/a<<1). Therefore the effective refractive index of the core mode is approximately equal to that of the core material n₁, n _eff =n₁cos(θ)≈n₁. As the wavelength that corresponding to the modal cutoff of the high-index inclusion, the effective refractive index of the highest order mode of the inclusion approaches n₁. Therefore at the cutoff wavelength of the high-index layer mode, the effective refractive index of the entire structure approaches n₁ as shown in Fig. 4. Upper plot in Fig. 4 shows the effective refractive index of the modes of the high-index layer. Lower plot corresponds to the effective refractive index of the low-index core mode of the entire ARROW structure. At a cutoff wavelength the entire waveguide structure becomes effectively transparent. Therefore, in order to find the wavelengths corresponding to the high loss regions for the entire ARROW structure, one needs to find modal cutoff conditions for the modes of a single high-index inclusion (layer or cylinder). For the planar structure this condition is equivalent to the resonant condition for the high-index layer given by Eq. (1). The modal properties of planar and cylindrical waveguide modes are not the same at the cutoff condition. Only some of the modes of the cylindrical step-index waveguide have zero power within the core at cutoff [27]. These include the TE_0m, TM_0m, HE_1m and HE_2m modes. Therefore, according to the Eq. (2) only these modes have a group velocity equal to the speed of light in the cladding c/n₁ (and n _eff =n₁). All other modes have some fraction of their power propagating in the core with the velocity smaller than c/n₁ (and n_eff>n₁). It seems reasonable that since only specific modes of the high-index inclusions have n _eff =n₁ at the cutoff, only those modes lead to appearance of the transmission minima of the entire structure. Therefore, in order to predict spectral minima of the entire MOF one needs to find modal cutoff wavelengths for TE_0m, TM_0m, HE_1m and HE_2m modes.

For TE_0m and TM_0m modal cutoff condition is given by [27]

For HE_1m mode modal cutoff is given by

Resonance conditions for k_t d/2 can be found analytically using the cosine approximation for the Bessel functions J_ν when k_t d/2>>ν.

From Eqs. (3) and (4) resonant wavelengths can be written as follows:

For HE_2m the cutoff condition is given by

The solution of transcendental Eq. (6) is very close to the solution of Eq. (3) given by Eq. (5) as long as

 

Fig. 4. Upper plot shows the effective refractive index of the modes of the high-index layer as a function of the wavelength. Lower plot shows the effective refractive index of the mode propagating in the low-index core of the entire ARROW structure. Vertical dashed lines correspond to the modal cutoffs.

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In our numerical example (shown in Fig. 3) $\frac{4\Delta 1}{1-2\Delta k_{t}d}=0.037÷0.055$. Therefore the condition (5) can be used to predict the high-loss spectral wavelengths in wide range of parameters. The analytical predictions of equation (5) are compared to numerical results obtained using the multipole method [18].

While closed-form analytical solutions for modal cutoff can be found in a limited number of cases such as a step-index planar or cylindrical waveguide, we can always find the cutoff wavelengths for other geometries of the high index inclusions numerically. Therefore, using the present approach spectral minima of any microstructured waveguide with low index core and high index inclusions of various shapes can be found, as long as these inclusions form waveguides.

In summary, it has been shown that the positions of spectral minima can be determined by calculating the modal cutoff wavelength of the high-index regions, and thus they depend only on the mode structure of those inclusions. Equations (1) and (5) provide good predictions for the positions of transmission minima (confinement loss maxima). Therefore, the complex problem of analyzing the properties of microstructured optical waveguides can be reduced to the much simpler problem of analyzing the modal cutoff properties of an individual high-index inclusion.

3. Potential applications

As shown in Fig. 2, the spectrum of an ARROW waveguide consists of several narrow transmission dips at wavelengths λ_m . These spectral dips can be utilized for making tunable optical filters. Since the resonant wavelengths in Eqs. (1) and (5) are functions of n₂, the position of the spectral dips can be shifted by changing n₂. Experimentally this can be realized by using materials with temperature dependent refractive index [13,14]. Resonant wavelength shift in planar and cylindrical geometries is given by

respectively.

In Eq. (8) we assume that the temperature sensitivity of a high-index material n2 is much higher than that of the background material n1. However, Eq. (8) can be easily modified to include the temperature dependence on n₁.

Let’s consider designing a tunable filter based on ARROW-like planar waveguide or MOF. First we design a continuously tunable filter based on the planar waveguide shown in Fig. 2.

 

Fig. 5. (a) Transmission minimum (m=10) for different values of n₂, for fixed values of n₁=1.4, d=3.437µm. (b) Comparison of the analytical predictions and the numerical simulations for the location of the transmission minimum.

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Fig. 6. (a) Schematic of MOF with a micro-heater. MOF air-holes are filled with a high-index material whose refractive index n2 changes with temperature T, (b) Longitudinal component of Poynting vector Sz for the lowest order MOF mode in transmission mode (n=1.8) and in filter mode (n=1.775) along with x cross section of Sz.

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We choose a particular transmission dip at λ=0.777µm,corresponding to m=10 in Eq. (1). Figure 5(a) shows the evolution of this peak over the wavelength as n₂ decreases. Comparison of the numerical results and analytical predictions for the location of the minimum versus n₂ is shown in Fig. 5(b). Note, that even though ARROW-type of waveguide is inherently lossy, that should not preclude this waveguide from being used as a filter since only a short piece of fiber is required for this application. Alternatively, additional high-index layers can be used to reduce the confinement loss.

A type of filter described here can be realized in MOF geometry by placing a micro-heater on a surface of MOF filled with thermally tunable material such as a high index liquid (n_{589 nm} =1.8,Cargille Laboratories series M with a temperature dependence of the refractive index given by dn/dT=-6.8*10^-4/deg C). The schematic of the device is shown in Fig. 6(a). In this example we use Eq. (5) to find a value for the refractive index n₂ at which the MOF with one ring of high-index inclusions switches from a transmission mode at initial (room) temperature to a filtering mode for λ=0.7525µm when the temperature increases.

Here we assume that the refractive index n₂ decreases with temperature and equals 1.8 at room temperature. The refractive index required to tune the MOF was found to be 1.775. In this example other fiber parameters are n₁=1.44, d=3.8µm, Λ=8µm. Figure 6(b) shows the longitudinal component of the Poynting vector S_z and its cross section along x axis for the lowest-order mode for n₂(T₁=1.8 and n₂(T₂)=1.775 calculated using the multipole method [23,24]. However, multipole simulations only provide an information about a particular core mode and do not explicitly show where the power goes when it disappears from the core at a resonant wavelength. BPM provides more information about the light propagation and the results obtained from simulations using BPM can be directly compared with experiments.

We used a BPM to propagate a Gaussian beam in z direction in 1mm of ARROW-like MOF with two rings of inclusions shown in Fig. 7(a). A second ring of inclusions has been added to reduce the power leakage from the core. All other parameters are identical to those in Fig. 6. Figure 7(b) shows the transmission spectra for two cases: n₂=1.8 (red line) and n₂=1.775 (black line). Both spectra are taken after beam propagated along the fiber (in zdirection) at z=1mm. Simulations shown in Fig. 7(c) and 7(d) further illustrate the dynamic of light propagation in ARROW-like MOF at a design wavelength λ=0.7525µm for n₂=1.8 (off-resonance) and n₂=1.775 (on-resonance), respectively. At this wavelength light propagates in the low-index core when n₂=1.8, since λ=0.7525µm corresponds to a high transmission part of the spectrum. Animation in Fig. 7(d) clearly shows that light escapes from the entire structure at the same wavelength when n₂=1.775. These simulations confirm that the MOF switches from a high transmission mode to a filtering mode as the refractive index changes as predicted by Eq. (5).

 

Fig. 7. (a) MOF profile used in BPM simulations, (b) Transmission spectra at z=1 mm, (c) (1446 KB) Evolution of the beam profile in MOF with n₂=1.8 (the frame shows output beam profile at z=1 mm), (d) (1446 KB) Evolution of the beam profile in MOF with n₂=1.775 (the frame shows output beam profile at z=1 mm).

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4. Summary

We reviewed some simple analytical approaches developed for understanding the spectral properties of MOFs with low refractive index core surrounded by high refractive index inclusions. Using the insight gained from a simple physical picture of light confinement and propagation in these MOFs, we demonstrated how the properties of these fibers can be adjusted in a controlled and predictable way providing a basis for novel photonic devices.