Abstract

The depressed core fiber (DCF), consisting of a low-index solid core, a high-index cladding and air surrounding, is in effect a bridge between the conventional step-index fiber and the tube-type hollow-core fiber from the point of view of the index profile. In this paper the dispersion diagram of a DCF is obtained by solving the full-vector eigenvalue equations and analyzed using the theory of anti-resonant and the inhibited coupling mechanisms. While light propagation in tube-type hollow-core fibers is commonly described by the symmetric planar waveguide model, here we propose an asymmetric planar waveguide for the DCFs in an anti-resonant reflecting optical waveguide (ARROW) model. It is found that the anti-resonant core modes in the DCFs have real effective indices, compared to the anti-resonant core modes with complex effective indices in the tube-type hollow-core fibers. The anti-resonant core modes in the DCFs exhibit similar qualitative and quantitative behavior as the core modes in the conventional step-index fibers. The full-vector analytical results for the simple-structure DCFs can contribute to a better understanding of the anti-resonant and inhibited coupling guidance mechanisms in other complex inversed index fibers.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Inversed index fibers with a low-index core and a high-index cladding such as the tube-type hollow-core fibers and hollow-core photonic crystal fibers have attracted a lot of interest in the fields of high-capacity telecommunication networks [15], high -power/supercontinuum/ultrafast lasers [611], terahertz waveguiding [12,13] and high sensitivity optical sensing [14,15]. As opposed to conventional step-index fibers where the light is guided in the high-index core region by total internal reflections, light guidance in the low-index core region of the inversed index fibers can be explained by other mechanisms such as the photonic bandgap effect, inhibited coupling effect and the anti-resonant effect. Generally, all optical fibers can be grouped into two types based on the effective refractive index (neff) -wavelength (λ) dispersion diagram [2]. For the first type, the neff - λ of the core modes lies outside any cladding mode continuum. The photonic bandgap fibers belong to this type, as their photonic bandgaps are formed in the periodic dielectric cladding region, the coupling between the core modes, which have neff within the bandgaps, and cladding modes are forbidden due to their separation in the neff - λ space [3]. The second type is inhibited coupling fibers, for which the neff - λ of the core modes lies inside the cladding mode continuum, but the coupling between them is minimized due to the high degree of transverse-field mismatch [6]. Both the photonic bandgap fibers and the inhibited coupling fibers have the same anti-resonant nature, which can be described by an anti-resonant reflecting optical waveguide (ARROW) model [2,3,6]. In this model, the planar waveguide acts as a Fabry-Perot resonator, which allows the anti-resonant light to be reflected back while allowing forward transmission of the resonant light [1619].

In a manner similar to photonic bandgap fibers, conventional step-index fibers belong to the first type since the neff of the core modes are higher than that of the cladding modes, in other words, their modal dispersion space is separated [2]. Compared to conventional step-index fibers, inversed index fibers have generally a more complex structure and modal characteristics. However, there exists a close relationship between them. It has been stated that the photonic bandgap fibers exhibit strikingly similar modal behavior with that of the conventional fibers, including dispersion curves and field profiles [20,21]. The analogy to conventional fibers provides a convenient tool to model the modes of photonic bandgap fibers. The question is whether there is a connection between the inhibited coupling fibers and the conventional fibers in addition to the photonic bandgap fibers.

A recent publication Ref. [22] shows that the mode density of anti-resonant elements in an inhibited coupling guiding single-ring hollow-core photonic crystal fiber is similar to the behavior of conventional multimode fibers. The single-ring hollow-core photonic crystal fiber studied in [22] has a ring of 6-8 detached thin tube-type hollow-core waveguides/fibers surrounding the hollow core. However, the results of the work were obtained using a scalar semi-analytical model, which does not give sufficiently rigorous vector modal analysis. The tube-type hollow-core fiber has a simple structure, where the leaky core-type modes are supported by the anti-resonant effect [1719]. In most of the previous works, only the fundamental mode or few low-order modes of the tube-type hollow-core fibers were obtained with different approximation methods [17,2325].

A depressed core fiber (DCF) is a type of inversed index fibers, described by a three-layer fiber model with a low-index solid core, a finite high-index cladding, and surrounding air. This fiber structure is different from the M-type fiber [26], although in some papers it has been referred to as M-type fiber [27]. The refractive index of the central region of the M-type fiber has a minimum value set by the need for the core to have a refractive index equal to or lower than that of the surrounding medium, which if the surrounding region is air means that the core must also be air. For a DCF the core refractive index is lower than the next outermost region but does not need to take on the lowest or minimum value in a three-layer fiber model. The M-type fiber can be considered as a leaky waveguide with the anti-resonant structure, in which the core mode is leaky with a complex effective refractive index, similar to the case of the hollow-core fiber [17,28]. Compared to the M-type fiber, the core mode of the DCF is guided by both total internal reflection and anti-resonance guidance, and its core mode has a real effective refractive index [27,2932]. The DCFs have been studied for various applications including pulse compression in fiber lasers [2931], generation of supercontinuum [27] and top-hat beams [32], due to their manageable waveguide dispersion and exceptional modal field changes. However, these studies in regard to DCFs [27,2932] were limited to the analysis of a few modes in the strong dispersion region.

In this paper, the modal dispersion and field distributions of a DCF are calculated and analyzed with the vector field functions and eigenvalue equations for the three-layer fiber as given in [3335]. To the best of our knowledge, it is the first report providing a complete vector modal dispersion diagram for the DCFs. In the section 2, the ray method is used to analyze the mode characteristics in the DCFs, in comparison to asymmetric planar waveguides. The possibility of anti-resonant core modes and the positions of the mode coupling (resonances) is analyzed using the ARROW model. In section 3, the dispersion curves and mode field distributions of the vector modes TE, TM, HE and EH are analyzed, along with their comparison for the case of conventional step-index fibers. It is found that the waveguiding mechanism of core-type modes in DCFs with a simple structure can be explained by the anti-resonant and inhibited coupling effects, similarly to the complex single-ring hollow-core photonic crystal fibers discussed in Ref. [22]. In section 4, the implications of the analytical results for the tube-type hollow-core fibers and single-ring hollow-core fibers are discussed.

2. Theory of the DCFs

Anti-resonant core mode and asymmetric planar ARROW model

Figure 1 shows a DCF and its cross section and refractive index profile. The radii of the rod core and the tube cladding are r1 and r2 while the surrounding medium (air) is unlimited. The refractive index of the rod core region n1 is smaller than the refractive index of the tube cladding region n2 but greater than that of the surrounding air n3, as n3 < n1 < n2.

 figure: Fig. 1.

Fig. 1. A depressed-core optical fiber and its cross section and refractive index profile.

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The high-index tube cladding region of the DCF can be considered as an asymmetric planar waveguide, where a high-index core region with the same thickness (d = r2 - r1) as the cladding of the DCF is sandwiched between two different low-index regions, as shown in Fig. 2. It is useful to analyze the DCFs using a ray optics approach. Since the asymmetric planar waveguide supports guided modes with n1 < neff = n2*sinθ1 < n2 [36] shown in Fig. 2(b), it can be deduced that a DCF can support the annular-like (transverse field profile) cladding modes with n1 < neff = n2*sinθ1 < n2 guided by the total internal reflections at the inner and outer boundaries of the cladding region as shown in Fig. 2(a). The modes with n3 < neff = n1*sinθ1 < n1 in the asymmetric planar waveguide are radiation modes with power escaping into the higher-index (upper) region (where will form a standing wave field) [36], as shown in Fig. 2(d). In the DCF an incident ray representing a mode with n3 < neff = n1*sinθ1 < n1, partially reflected at the inner boundary of the cladding region and totally reflected at the outer boundary of the cladding region, excites multiple path rays reflecting into the core region. The light field of these modes with n3 < neff < n1 is distributed in across the entire cross section of the DCF, which is similar to the cladding modes in conventional three-layer step-index fibers [37,38].

 figure: Fig. 2.

Fig. 2. (a) and (b) show the ray trajectory of a mode with n1 < neff = n2sinθ1 < n2 in a DCF and an equivalent asymmetric planar waveguide, respectively. (c) and (d) show the ray trajectory of a mode with n3 < neff = n1sinθ1 < n1 in the DCF and the equivalent asymmetric planar waveguide, respectively.

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Due to the inversed index distribution in the DCF, the total internal reflection effect does not work at the inner boundary of the cladding region. Therefore, there is no core modes guided by the total internal reflections like those in conventional step-index fibers.

The high-index layers can be considered as Fabry-Perot resonators in the ARROW model [16]. Analogous to a symmetric planar waveguide in the ARROW model for the tube-type hollow-core fibers [18,19], an asymmetric planar waveguide is proposed for the DCFs. Indeed, the characteristics of multiple path reflections formed in the DCF shown in Fig. 2(c) indicate that the cladding region of the DCF acts as a Fabry-Perot resonator, corresponding to an asymmetric planar waveguide shown in Fig. 2(d). In a Fabry-Perot resonator the resonances usually occur over a narrow band of wavelengths while the antiresonances are spectrally broad [16,39]. The energy of the incident ray can be strongly reflected back to the core region at the anti-resonant wavelengths, forming anti-resonant core modes in the anti-resonant reflecting optical waveguides [16] and the tube-type hollow core fibers [18]. Similarly, it can be concluded that DCFs have anti-resonant core modes with the light field is mainly confined in the core region, with disc-like transverse field profiles.

The resonant wavelengths of an asymmetric planar waveguide-like Fabry-Perot resonator shown in Fig. 2(d), corresponding to the minimal total energy of all the reflected rays, are equal to the cutoff wavelengths of the guided core modes. The cutoff wavelengths for the guided core modes TEN and TMN in the asymmetric waveguide can be written as [36]:

$${\lambda _{\textrm{N},\textrm{c}}} = \frac{{2d\sqrt {n_2^2 - n_1^2} }}{{\left[ {\textrm{N} + \frac{1}{\pi }{{\tan }^{ - 1}}\left( {\kappa \frac{{\sqrt {n_1^2 - n_3^2} }}{{\sqrt {n_2^2 - n_1^2} }}} \right)} \right]}},\; \; \kappa = \left\{ {\begin{array}{{c}} {1,\; \; \; \; {\textrm{for}}\; \textrm{T}{\textrm{E}_\textrm{N}}\; {\textrm{modes}}\; }\\ {\frac{{n_2^2}}{{n_3^2}}\; \; \; {\textrm{for}}\; \textrm{T}{\textrm{M}_\textrm{N}}\; {\textrm{modes}}} \end{array}} \right.$$
where N is the mode number. The second term in the denominator of Eq. (1) arises from the asymmetry of the waveguide and vanishes for the symmetric case as shown in Ref. [19].

3. Results

Similar to the conventional step-index fibers, the modes in a DCF are denoted as TE0,N, TM0,N, HEm,N and EHm,N, where the numbers ‘0’ and ‘m’ on the left side of the comma in the subscript positions are the azimuthal mode number while the number symbols ‘N’ on the right side are the radial mode number. The eigenvalue equations for the vector modes in the DCFs are shown in Appendix, which were solved by a graphical method. In the calculations, the fiber parameters of the studied DCF are assumed as n1 = 1.445, n2 = 1.51, n3 = 1, r1 = 62.5 µm and r2 = 125 µm. The parameters correspond to those of the fibers used in our experimental work, which will soon be presented in another paper.

3.1 TE/TM modes in DCFs

Figure 3(a) shows the dispersion diagram (neff vs. λ) for the modes TE0,N with n1 < neff < n2. There are 38 TE0,N modes, from upper to lower as the radial mode number N increases, as indicated by the black arrow. All dispersion curves of the TE0,N modes with the purple dashed line at n1, but only the modes TE0,N, N = 34, 35, 36, 37 and 38 have their cutoff in the investigated spectral range. The wavelengths of the intersect points are approximated as the cutoff wavelengths of the TEN modes in the asymmetric planar waveguide calculated by Eq. (1), indicated by the red vertical dashed lines. The difference between the wavelength of the intersect point for the TE0,N mode and the cutoff wavelength of the TEN mode is less than 1 nm. Compared to the curves above the horizontal line at n1, the curves bellow n1 show a step-like decrease as the λ increases, showing periodic strong and moderate index dispersion bands. The slopes of the dispersion curves with a strong index dispersion are similar to those of the curves above the horizontal line at n1 and that in the equivalent asymmetric waveguide (not shown). The slopes of the dispersion curves with a moderate index dispersion in DCFs are similar to those of the dispersion curves of the core modes in the conventional step-index fiber, as discussed in part 3.3.

 figure: Fig. 3.

Fig. 3. Dispersion curves (neff vs. λ) of TE modes with neff corresponding to (a) n1 < neff < n2, (b) n1-0.001 < neff < n1+0.001, n1 = 1.445, n2 = 1.51. (c) shows the modal intensity and electric field vector distributions of TE modes whose positions (neff, λ) are indicated by the black circle dots in (a) and (b). The red vertical dashed lines in (a) and (b) indicate the resonant bands.

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Figure 3(c) shows the modal intensities and electric field vector distributions at seven different points marked A-G (black circle dots) in Figs. 3(a) and 3(b). The electric field vector helps to distinguish between different kinds of vector modes and helps to compare the modes in the DCF and the conventional step-index fiber, as discussed in following parts. The radial maxima in the modal field profiles represent intensity oscillations, and the number of the radial maxima is equal to the radial mode number N. In Fig. 3(c) the modal field profile of the mode TE0,1 at point A shows one radial maxima in the fiber cladding region r1 < r < r2. The TE0,2 at point B shows two radial maxima in the fiber cladding region.

In order to count the number of radial maxima in the distributions corresponding to points C-G in Fig. 3(c), we draw the normalized intensity distribution along the r-coordinate, as shown in Fig. 4. One oscillation peak on a curve in Fig. 4 corresponds to one radial maxima in a modal field profile. Similar to the modal field profiles at points A and B, all 37 radial maxima are distributed within the fiber cladding region for the mode at point C. The oscillation peaks shift from the cladding region to core region one by one along the points C-G. The modes at points D, F and G exhibit 1, 2 and 3 maxima in the core and 36, 35 and 34 maxima in the cladding, respectively. The energy is mainly confined in the core region of the modes at points D, F and G, as seen in Figs. 4(b), 4(d) and 4(e), where the field intensity in the cladding region is almost negligible compared to that in the core region. The mode at point E with a very strong index dispersion represents a mode in a transition state from the mode at point D to the mode at point F, where the energy is more evenly distributed between the core and the cladding regions as seen in Fig. 4(c). There are two maxima in the core region and 35 maxima in the cladding region for the mode at point E.

 figure: Fig. 4.

Fig. 4. The left panel (a)-(e) shows the normalized intensity distributions along the r-coordinate for the modes shown at points C-G in Fig. 3. The right panel (a′)-(e′) shows the zoomed-in part of (a)-(e), delineated by the vertical dashed lines.

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We propose the following nomenclature for the modes with neff < n1, identified as TE0, n+(N-n). The subscript ‘n’ denotes the radial number in the core region, while the number ‘N-n’ denotes the radial number in the cladding region. Therefore, the modes at points D, F and G in Fig. 3(b) with moderate index dispersion can be named as TE0,1+(36), TE0,2+(35) and TE0,3+(34). The mode at point E in Fig. 3(b) can be named as [TE0,2+(35)], where the brackets indicate that the mode is with a strong effective index dispersion and is in a transition state. It is clear that the field profiles of the modes TE0,1+(36), TE0,2+(35) and TE0,3+(34) are similar to that of the modes TE0,1, TE0,2 and TE0,3 in conventional step-index fibers, respectively. As discussed, the TE0,1+(36), TE0,2+(35) and TE0,3+(34) in the DCF should be formed by the anti-resonant effect of the high-index cladding region, therefore, they represent anti-resonant core modes.

According the above nomenclature, the modes at points I and J on the dispersion curve of TE0,36 indicated by the black triangle dots in Fig. 3(b) are both anti-resonant core modes and named as TE0,1+(35) and TE0,2+(34), respectively. The mode transition from the core mode TE0,1+(35) at point I to the cladding mode TE0,36 at point H and the mode transition from the higher-order core mode TE0,2+(35) at point F to the lower-order core mode TE0,1+(36) at point D leads to an anti-crossing phenomenon near the mode [TE0,2+(35)] at point E around the resonant bands indicated by the red vertical dashed line. Due to mode reorganization [38,4042], the anti-crossing phenomenon also takes place for the higher order modes. A similar phenomenon is also observed for the TM, HE and EH modes. The anti-crossing phenomenon originates from the resonant coupling between the core and the cladding modes, leading to the exceptional waveguide dispersion and modal field changes. As shown in Figs. 3 and 4 the modal field distribution and effective refractive index dispersion change drastically from the points D or F to the point E. The mode [TE0,2+(35)] at point E show similar strong dispersion with the cladding modes in the DCF and the modes in the equivalent asymmetric waveguide. Therefore, the wavelength position and the dispersion shape of the anti-crossing are related to the thickness and the refractive index of the high-index cladding, in accordance with the ARROW model and Eq. (1) as discussed in the theory section. Due to the manageable waveguide dispersion and modal field changes in a DCF, the generation of supercontinuum [27] and top-hat beams [32] can be achieved.

The dispersion diagram of TM0,N modes is similar to that of TE0,N modes, as seen in Fig. 5. The positions of the intersect points between the dispersion curves of the TM0,N and the horizontal line at n1 can be approximated as the cutoff wavelengths of the TMN modes in the asymmetric planar waveguide calculated by Eq. (1), indicated by the black vertical dashed lines in Figs. 5(a) and 5(b). Figure 5(c) shows the modal intensity and electric field vector distributions of TM modes at points A'-G’ indicated by the black squares in Figs. 5(a) and 5(b). The modes at points A’ and B’ are TM0,1 and TM0,2, respectively. The points C'-G’ are on the dispersion curve of TM0,37. The field intensity distributions of TM modes in Fig. 5(c) are similar with those of TE modes shown in Fig. 3(c). However, the directions of the electric field vector of TM and TE modes are different: the former is parallel to the radial direction while the latter is normal in the radial direction.

 figure: Fig. 5.

Fig. 5. Dispersion curves (neff vs. λ) of TM modes with neff corresponding to (a) n1 < neff < n2, (b) n1-0.001 < neff < n1+0.001, n1 = 1.445, n2 = 1.51. (c) shows the modal intensity and electric field vector distributions of TM modes whose positions (neff, λ) are indicated by the black squares in (a) and (b). The black vertical dashed lines in (a) and (b) indicate the resonant bands.

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Similarly to the nomenclature of the TE modes, the TM modes with neff < n1 can be named as TM0,n+(N-n). Therefore, the anti-resonant core modes at points D’, F’ and G’ in Fig. 5(b) with a moderate effective index dispersion can be named as TM0,1+(36), TM0,2+(35) and TM0,3+(34), respectively. The mode at point E’ in Fig. 5(b) can be named as [TM0,2+(35)]. From the modal field distributions, it is clear that the anti-resonant core modes TM0,1+(36), TM0,2+(35) and TM0,3+(34) are similar to the core modes TM0,1, TM0,2 and TM0,3 in the conventional step-index fibers, respectively.

3.2 HE/EH modes in DCFs

The HE or EH in the DCFs are two-fold degenerate modes with the same effective refractive index but different field vector directions, similar to the HE/EH modes in conventional step-index fibers. Here we only show the results for one of these two-fold degenerate modes.

Figure 6 illustrates the dispersion curves of the modes HE1,N and EH1,N (N = 34, 35,…, 38). Similar with the TE and TM modes in Figs. 3(b) and 5(b), the neff of HE1,N and EH1,N decrease linearly with the increase of λ when neff > n1 while bellow n1 the neff decrease in a step-like fashion as λ increases. The curves show periodic strong and moderate index dispersion bands. The positions of the intersect points between the dispersion curves of modes HE1,N (EH1,N) and the horizontal line at n1 can be approximated as the cutoff wavelengths of the TEN (TMN) modes in the asymmetric planar waveguide calculated by Eq. (1), indicated by the red (black) vertical dashed lines in Fig. 6.

 figure: Fig. 6.

Fig. 6. Dispersion curves (neff vs. λ) of HE1,N (red) and EH1,N (black) modes with neff corresponding to (n1-0.001 < neff < n1+0.001, n1 = 1.445. The black and red vertical dashed lines indicate the resonant bands.

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For HEm,N or EHm,N modes, each radial mode order ‘N’ supports a larger number of azimuthal mode orders ‘m’. Figure 7(a) as an example shows a dispersion diagram of the HEm,N modes with m = 1, 2, … and N = 34, 35, …, 38. The curves with the same radial mode order ‘N’ are shown in the same color. For the sake of clarity, the transparency of the curves has been increased toward higher azimuthal mode orders in each group. All these curves form a very dense dispersion diagram, where the curves with moderate slopes intersect with those having steeper slopes.

 figure: Fig. 7.

Fig. 7. Dispersion curves (neff vs. λ) of HE modes with neff corresponding to (a) n1-0.001 < neff < n1+0.001, n1 = 1.445. (b) partially enlarged image of (a), indicated by the red frame. (c) shows the modal intensity and electric field vector distributions of HE modes whose positions (neff, λ) are indicated by the circles in (b). The black and red vertical dashed lines in (a) and (b) indicate the resonant bands.

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Figure 7(b) shows the zoomed view of the part indicated by a red rectangle in Fig. 7(a), showing the transition of modes HEm,37, m = 1, 2, …. The dispersion curve of the mode HEm,37 changing from a steep slope to a moderate slope indicates the formation of an anti-resonant core-type mode HEm,1+(36) (the nomenclature similar to TE/TM modes). For example, the modes HE1,37 at point H1 and HE3,37 at point H3 are transferred into the modes HE1,1+(36) at point H2 and HE3,1+(36) at point H4, respectively. Their modal field intensity and electric vector distributions are shown in Fig. 7(c). As the azimuthal mode order ‘m’ increases, the transition of the cladding-type HEm,37 to anti-resonant core-type HEm,1+(36) will occur at a longer wavelength and a smaller effective index at the point where the slope of the dispersion curve changes. Therefore, the transition process for the modes with large azimuthal mode orders ‘m’ may not be visible in the given range of the dispersion diagram. As an example, although the dispersion curve of the mode HE15,37 in Fig. 7(b) intersects the horizontal line corresponding to the value n1, it does not change direction abruptly, indicating no mode transition in the given range. Two modes at points H5 and H6 on this dispersion curve of the mode HE15,37 show similar modal field intensity and electric vector distributions as seen in Fig. 7(c). These two modes are both cladding-type modes with energy confined in the cladding region, however, their formation mechanisms are likely different. Compared to the cladding mode at point H5 with neff > n1 guided by the total internal reflections at both the inner and outer boundaries of the cladding region, the cladding mode at point H6 with neff < n1 cannot be guided by the total internal reflections at the inner boundary of the cladding region. The existence of the cladding modes in the dispersion space below the horizontal line at n1 in Figs. 7(a) and 7(b) is likely attributed to the inhibited coupling effects as per following discussion.

The modal field intensity and electric vector distributions of two anti-resonant core modes HE1,1+(37) at points H7 and H8 are drawn in Fig. 7(c), showing that for both of the modes the field energy is confined in the fiber core region. The intersection of the dispersion curve of HE1,1+(37) with that of HE15,37 indicates that the disc-like core modes and the annular-like cladding modes can exist in the same dispersion space (neff vs. λ). The same neff indicates that the modes have the same longitudinal components of the light field. However, they cannot couple with each other since their transverse components of the light field are different, corresponding to the inhibited coupling effect [6].

3.3 Mode degeneracy in DCFs

The dispersion curves of several lower order TE, TM, HE and EH modes in the DCF are drawn together in the same dispersion diagram, shown in Fig. 8(a). Other higher order modes in the DCF are not shown in order to preserve clarity of the diagram. The dispersion diagram shows periodic resonant and anti-resonant bands corresponding to the strong and moderate index dispersion, and the resonant bands are indicated by the black and red vertical dashed lines. Figure 8(b) displays one of such periodic bands, indicated by the black dashed rectangle. The dispersion curves of the anti-resonant core modes in the DCF such as {HE1,1+(36)}, {TE0,1+(36), HE2,1+(36) and TM0,1+(36)} and {EH1,1+(36) and HE3,1+(36)} can be respectively grouped together. In calculation, the longitudinal components of the light field are very small compared to the transverse components of the light field for the anti-resonant core modes. Therefore, the modes in each group can be grouped into a single degenerate scalar mode or a linear polarized mode, similar to that in the conventional step-index fibers. These degeneracies are broken in the resonant bands, where dispersion curves of TE, TM, HE and EH modes are separated.

 figure: Fig. 8.

Fig. 8. (a) Dispersion curves (neff vs. λ) of a depressed core fiber. (b) partially enlarged image of (a), indicated by the black dashed frame. The text labels and the corresponding dispersion curves are of the same color. The orange solid lines in (a) and (b) are for LPm,n modes in a conventional step-index fiber (nco = 1.445, ncl = 1, rco = 62.5 um and rcl = ∞). The black and red vertical dashed lines indicate the resonant bands.

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If the high-index cladding region is removed, the fiber becomes a conventional step-index fiber, consisting of a bare core and the surrounding air acting as the fiber cladding, where nco = 1.445, ncl = 1, rco = 62.5 um and rcl = ∞. The dispersion curves of the core modes in such a step-index fiber were calculated by graphical method. The vector modes in the obtained fiber such as {HE1,1}, {TE0,1, HE2,1 and TM0,1} and {EH1,1 and HE3,1} can be respectively grouped into linear polarization modes LP0,1, LP1,1 and LP2,1, due to their mode fields being far from cutoff [37,38]. Figure 8 shows the dispersion curves of modes LPm,n in the step-index fiber in orange color, which are overlapped with the moderate dispersion curves of the anti-resonant core modes in the DCF. The results indicate that the anti-resonant core modes {HE1,1+(36)}, {TE0,1+(36), HE2,1+(36) and TM0,1+(36)} and {EH1,1+(36) and HE3,1+(36)} in the DCF show similar qualitative and quantitative behaviour with the corresponding core modes {HE1,1}, {TE0,1, HE2,1 and TM0,1} and {EH1,1 and HE3,1} in the conventional step-index fiber. In addition, the effective refractive index of the anti-resonant core modes in the DCF can be approximated as that of the corresponding LPm,n modes in a conventional step-index fiber.

4. Discussion

The dispersion diagram of the DCF shows some similar characteristics with the dispersion diagram of a single-ring hollow-core anti-resonant fiber given in [22]. Firstly, they both show periodic resonant and anti-resonant bands. Secondly, they both show the inhibited coupling phenomenon between the core modes and the cladding modes. However, due to the complex structure of the single-ring hollow-core anti-resonant fibers, only an approximate scalar method was used in [22]. Using the full-vector analytical method in this paper, the transition between the cladding modes and the anti-resonant core modes has been demonstrated. The degeneracy in the anti-resonant bands and the loss of degeneracy around the resonant bands make the dispersion curves of anti-resonant core modes merge together or separate, which may correspond to the narrow and wide variations in bands of anti-resonant modes in [22].

Although the inhibited coupling fiber is different with the photonic bandgap fiber from the modal dispersion perspective, both fibers can be viewed as analogous to the conventional step-index fibers. Reference [20] shows that the density of core modes in the bandgap fibers is similar to that of the conventional step-index multimode fibers. Our work verified that the density of the anti-resonant core modes in the DCF (a inhibited coupling fiber) is equal to that of the core modes in the equivalent conventional step-index fiber, since the anti-resonant core modes in the former have a one-to-one correspondence to the core modes in the latter.

The core modes in both DCFs and tube-type hollow-core fibers are formed by the anti-resonant effect, yet their properties are different. The neff of the core modes in DCFs is higher than the refractive index of the surrounding air, therefore they are guided by total internal reflections at the outer boundary of the cladding region and they are non-leaky modes with a real neff. Compared to the DCFs, the core modes in a tube-type hollow-core fiber are usually treated as leaky modes with a complex neff. The leaky modes can be solved by the eigenvalue equations in the complex plane, but it is extremely cumbersome. In most of the previous works, only the fundamental mode or a few low-order modes of the hollow-core fibers were obtained with different approximation methods [17,2325]. However, the available results for the tube-type hollow-core fibers show some similar characteristics with those for the DCFs. Reference [17] reports step-like dispersion curves for the modes HE1,n+(N-n) (n = 1, 2, 3, 4, where the radial number ‘N-n’ in the cladding is not clear), which is similar to the behaviour of the HE1,n+(N-n) modes in the DCFs presented here. Therefore, it can be concluded that all other modes for the tube-type hollow-core fibers are similar to those in DCFs. Since the tube-type hollow-core fiber has a dispersion diagram similar to that of the DCF, the tube-type hollow-core fibers should be considered as inhibited coupling fibers.

The anti-resonance derived anti-crossing phenomenon takes place both in the DCF and in the hollow-core photonics crystal fibers (for either the photonic bandgap guiding or inhibited coupling guiding). The anti-crossing phenomenon has been intensively studied in hollow-core photonic crystal fibers for applications such as generation of the multi-octave supercontinuum [10], ultrafast nonlinear dynamics optics [11] and the broadband robustly single-mode guidance [12]. Therefore, given that DCFs can readily provide manageable waveguide dispersion and exceptional modal field changes, they are worthy further study for a wide range of applications, in addition to the few existing works such as the generation of the supercontinuum [27] and top-hat beams [32]. Furthermore high refractive index coated step-index fibers have been used for sensing of organic vapors, humidity, voltage, pH, and chemical/bio analytes [43] and have a similar refractive index profile to that of DCF which suggests the possibility that DCF might also be useful for sensing. Probably because the thickness of the coating in these structures is small, the inverted refractive index profile and the anti-resonant effect were not observed. Our ongoing works will explore the application of the anti-resonant and inhibited coupling characteristics of the DCFs in optical fiber sensing.

5. Conclusion

DCFs were studied analytically in comparison with the conventional step-index fibers and the tube-type hollow-core fibers and were found to be a form of anti-resonant and inhibited coupling fibers. In this work an asymmetric planar waveguide approach was proposed for the DCFs in the ARROW model. It has been shown that the DCFs support annular-like cladding modes in the tube cladding region and disc-like anti-resonant core modes in the rod core region and both of them were obtained by solving the same group of full-vector eigenvalue equations using the graphical method. The calculated dispersion diagram shows periodic resonant and anti-resonant bands, where the dispersion curves of the anti-resonant core modes intersect with those of the cladding modes. The formation of core-type modes in a low-index core region can be explained by both the anti-resonant and inhibited coupling mechanisms. The anti-resonant core modes exhibit similar qualitative and quantitative behaviors as those of the conventional step-index fibers. The analogy to conventional step-index fibers may provide a convenient tool to model the modes of the DCFs (the inhibited coupling fibers). To the best of our knowledge, it is the first report on the complete vector modal dispersion diagram calculated analytically for an inversed index fiber. Our results can be used to provide a better understanding of the anti-resonant and inhibited coupling guidance mechanisms in complex inversed index fibers such as hollow-core photonic-bandgap fibers, tube-type hollow-core fibers and single-ring hollow-core fibers.

Appendix

Eigenvalue equations for TE, TM, HE and EH modes in DCFs

From the analysis in Section 2, the modes with n3 < neff < n1 are like the cladding modes in three-layer step-index fibers. Therefore, the eigenvalue equations used to calculate the effective refractive index of the modes with n3 < neff < n1 in DCFs is the same as that for the cladding modes in the three-layer stepped-index fibers, as follows [33]:

for the TE0,N modes with n3 < neff < n1:

$$\hat{J}\left( {K{p_m} + \frac{{{r_m}}}{{{\alpha_2}{U_2}}}} \right) = \frac{1}{{{U_2}}}\left( {\hat{K}{q_m} + \frac{{{s_m}}}{{{\alpha_2}{U_2}}}} \right),\; \textrm{m} = 0$$
for the TM0,N modes with n3 < neff < n1:
$$\hat{J}\left( {\hat{K}{p_m} + {s_{23}}\frac{{{r_m}}}{{{\alpha_2}{U_2}}}} \right) = \frac{{{s_{21}}}}{{{U_2}}}\left( {\hat{K}{q_m} + {s_{23}}\frac{{{s_m}}}{{{\alpha_2}{U_2}}}} \right),\; \textrm{m} = 0$$
for the HEm,N and the EHm,N modes with n3 < neff < n1:
$$\begin{array}{l} p_m^2 + 2{\left( {\frac{2}{{\pi {\alpha_2}U_2^2}}} \right)^2}\left( {\frac{{n_2^2}}{{{n_1}{n_3}}}} \right){x_1}{x_2}\\ + x_1^2x_2^2\left[ {\hat{J}\left( {\hat{K}{p_m} + \frac{{{r_m}}}{{{\alpha_2}{U_2}}}} \right) - \frac{1}{{{U_2}}}\left( {\hat{K}{q_m} + \frac{{{s_m}}}{{{\alpha_2}{U_2}}}} \right)} \right]\\ \times \left[ {\hat{J}\left( {\hat{K}{p_m} + {s_{23}}\frac{{{r_m}}}{{{\alpha_2}{U_2}}}} \right) - \frac{{{s_{21}}}}{{{U_2}}}\left( {\hat{K}{q_m} + {s_{23}}\frac{{{s_m}}}{{{\alpha_2}{U_2}}}} \right)} \right]\\ = x_1^2\left( {\hat{J}{p_m} - \frac{{{q_m}}}{{{U_2}}}} \right)\left( {\hat{J}{p_m} - {s_{21}}\frac{{{q_m}}}{{{U_2}}}} \right) + x_2^2\left( {\hat{K}{p_m} + \frac{{{r_m}}}{{{\alpha_2}{U_2}}}} \right)\left( {\hat{K}{p_m} + {s_{23}}\frac{{{r_m}}}{{{\alpha_2}{U_2}}}} \right) \end{array}$$
The fiber parameters of the DCF shown in Fig. 1 are r1, r2, n1, n2 and n3. The wavenumber in vacuum is: ${k_0} = 2\pi /\lambda $, where λ is the wavelength. The longitudinal propagation constant is: $\beta = {k_0}{n_{\textrm{eff}}}$. In the Eqs. (2), (3) and (4), the parameters used are as follows:
$${\alpha _2} = \frac{{{r_2}}}{{{r_1}}},$$
$${u_1} = \sqrt {k_0^2n_1^2 - {\beta ^2}} \; ,\; $$
$$\; {u_2} = \sqrt {k_0^2n_2^2 - {\beta ^2}} ,\; $$
$$\; \; {\omega _3} = \sqrt {{\beta ^2} - k_0^2n_3^2} \; ,$$
$${U_1} = {u_1}{r_1}\; ,\; \; {U_2} = {u_2}{r_1}\; ,\; \; {W_3} = {\omega _3}{r_2},\; $$
$$\hat{J} = \; \frac{{J'_m({{U_1}} )}}{{{U_1}{J_m}({{U_1}} )}}\; ,$$
$$\; \hat{K} = \; \frac{{K'_m({{W_3}} )}}{{{W_3}{K_m}({{W_3}} )}},$$
$${p_m} = {J_m}({{u_2}{r_2}} ){Y_m}({{u_2}{r_1}} )- {J_m}({{u_2}{r_1}} ){Y_m}({{u_2}{r_2}} ),$$
$${q_m} = {J_m}({{u_2}{r_2}} )Y'_m({{u_2}{r_1}} )- J'_m({{u_2}{r_1}} ){Y_m}({{u_2}{r_2}} ),$$
$${r_m} = J'_m({{u_2}{r_2}} ){Y_m}({{u_2}{r_1}} )- {J_m}({{u_2}{r_1}} )Y'_m({{u_2}{r_2}} ),$$
$${s_n} = J'_m({{u_2}{r_2}} )Y'_m({{u_2}{r_1}} )- J'_m({{u_2}{r_1}} )Y'_m({{u_2}{r_2}} ),$$
$${s_{23}} = \frac{{n_2^2}}{{n_3^2}},\; {s_{21}} = \frac{{n_2^2}}{{n_1^2}},\; \; $$
$$V_{12}^2 = k_0^2r_1^2({n_1^2 - n_2^2} ),\; V_{23}^2 = k_0^2r_2^2({n_2^2 - n_3^2} ),$$
$$x_1^2 = \frac{{n_1^2U_1^4U_2^4}}{{\sigma _0^2V_{12}^4}},\; x_2^2 = \frac{{n_3^2\alpha _2^4U_2^4W_3^4}}{{\sigma _0^2V_{23}^4}},$$
$$\sigma _0^2 = {\left( {\frac{{\beta m}}{{{k_0}}}} \right)^2}.$$
The parameters ${u_1},\; \; {u_2},\; \; {\omega _3},\; {U_1},\; {U_2},\; \; \; {W_3}$ in Eqs. (6)-(9) are phase parameters. The functions ${J_m}$, ${Y_m}$ and ${K_m}$ in Eqs. (10)-(15) denote the Bessel function of the first kind, the Bessel function of the second kind and the modified Bessel function of the second kind. $J'_m$, $Y'_m$, and $K'_m$denote the derivatives of the corresponding Bessel functions.

To calculate the TE, TM, HE and EH modes with n1 < neff < n2, the phase parameter u1 in the Eq. (6) needs to be modified as:

$${u_1} = \sqrt {{\beta ^2} - k_0^2n_1^2} \; $$
and $\hat{J}$ in the Eq. (10) need to be modified as:
$$\hat{J} = \; - \frac{{I'_m({{U_1}} )}}{{{U_1}{I_m}({{U_1}} )}},\; \; $$
where ${I_m}$ and $I'_m$ denote the modified Bessel function of the first kind and its derivative.

The modal intensity and electric field vector distributions can be calculated with the field functions shown in Refs. [3335].

Funding

Technological University Dublin (FIOSRAIGH 2016 scholarship program).

Acknowledgments

We thank Dr. Cian Taylor from the Institute of Technology Sligo for valuable discussions.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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References

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  1. S. F. Gao, Y. Y. Wang, W. Ding, D. L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9(1), 1–6 (2018).
    [Crossref]
  2. B. Debord, A. Amsanpally, M. Chafer, A. Baz, M. Maurel, J. M. Blondy, E. Hugonnot, F. Scol, L. Vincetti, F. Gérôme, and F. Benabid, “Ultralow transmission loss in inhibited-coupling guiding hollow fibers,” Optica 4(2), 209–217 (2017).
    [Crossref]
  3. F. Poletti, M. N. Petrovich, and D. J. Richardson, “Hollow-core photonic bandgap fibers: technology and applications,” Nanophotonics 2(5-6), 315–340 (2013).
    [Crossref]
  4. F. Yu and J. C. Knight, “Negative curvature hollow-core optical fiber,” IEEE J. Sel. Top. Quantum Electron. 22(2), 4400610 (2016).
    [Crossref]
  5. Y. Chen, Z. Liu, S. R. Sandoghchi, G. T. Jasion, T. D. Bradley, E. N. Fokoua, J. R. Hayes, N. V. Wheeler, D. R. Gray, B. J. Mangan, R. Slavík, S. Member, F. Poletti, M. N. Petrovich, S. Member, and D. J. Richardson, “Multi - kilometer long, longitudinally uniform hollow core photonic bandgap fibers for broadband low latency data transmission,” J. Lightwave Technol. 34(1), 104–113 (2016).
    [Crossref]
  6. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318(5853), 1118–1121 (2007).
    [Crossref]
  7. A. F. Kosolapov, A. D. Pryamikov, A. S. Biriukov, V. S. Shiryaev, M. S. Astapovich, G. E. Snopatin, V. G. Plotnichenko, M. F. Churbanov, and E. M. Dianov, “Demonstration of CO2-laser power delivery through chalcogenide-glass fiber with negative-curvature hollow core,” Opt. Express 19(25), 25723–25728 (2011).
    [Crossref]
  8. A. D. Pryamikov, A. S. Biriukov, A. F. Kosolapov, V. G. Plotnichenko, S. L. Semjonov, and E. M. Dianov, “Demonstration of a waveguide regime for a silica hollow–core microstructured optical fiber with a negative curvature of the core boundary in the spectral region > 3.5 µm,” Opt. Express 19(2), 1441–1448 (2011).
    [Crossref]
  9. P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
    [Crossref]
  10. R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
    [Crossref]
  11. F. Tani, F. Köttig, D. Novoa, R. Keding, and P. S. Russell, “Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonic crystal fibers,” Photonics Res. 6(2), 84–88 (2018).
    [Crossref]
  12. P. Uebel, M. C. Günendi, M. H. Frosz, G. Ahmed, N. N. Edavalath, J.-M. Ménard, and P. S. J. Russell, “Broadband robustly single-mode hollow-core PCF by resonant filtering of higher-order modes,” Opt. Lett. 41(9), 1961–1964 (2016).
    [Crossref]
  13. C.-H. Lai, B. You, J.-Y. Lu, T.-A. Liu, J.-L. Peng, C.-K. Sun, and H.-C. Chang, “Modal characteristics of antiresonant reflecting pipe waveguides for terahertz waveguiding,” Opt. Express 18(1), 309–322 (2010).
    [Crossref]
  14. S. Liu, Y. Ji, L. Cui, W. Sun, J. Yang, and H. Li, “Humidity-insensitive temperature sensor based on a quartz capillary anti-resonant reflection optical waveguide,” Opt. Express 25(16), 18929–18939 (2017).
    [Crossref]
  15. D. Liu, Q. Wu, C. Mei, J. Yuan, X. Xin, A. K. Mallik, F. Wei, W. Han, R. Kumar, C. Yu, S. Wan, X. He, B. Liu, G.-D. Peng, Y. Semenova, and G. Farrell, “Hollow Core Fiber Based Interferometer for High Temperature (1000 °C) Measurement,” J. Lightwave Technol. 36(9), 1583–1590 (2018).
    [Crossref]
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2019 (4)

2018 (4)

S. S. Aleshkina, M. V. Yashkov, M. M. Bubnov, A. N. Guryanov, and M. E. Likhachev, “Asymptotically single-mode hybrid fiber for dispersion management near 1 µm,” IEEE J. Sel. Topics Quantum Electron 24(3), 1–8 (2018).
[Crossref]

S. F. Gao, Y. Y. Wang, W. Ding, D. L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9(1), 1–6 (2018).
[Crossref]

F. Tani, F. Köttig, D. Novoa, R. Keding, and P. S. Russell, “Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonic crystal fibers,” Photonics Res. 6(2), 84–88 (2018).
[Crossref]

D. Liu, Q. Wu, C. Mei, J. Yuan, X. Xin, A. K. Mallik, F. Wei, W. Han, R. Kumar, C. Yu, S. Wan, X. He, B. Liu, G.-D. Peng, Y. Semenova, and G. Farrell, “Hollow Core Fiber Based Interferometer for High Temperature (1000 °C) Measurement,” J. Lightwave Technol. 36(9), 1583–1590 (2018).
[Crossref]

2017 (5)

S. Liu, Y. Ji, L. Cui, W. Sun, J. Yang, and H. Li, “Humidity-insensitive temperature sensor based on a quartz capillary anti-resonant reflection optical waveguide,” Opt. Express 25(16), 18929–18939 (2017).
[Crossref]

B. Debord, A. Amsanpally, M. Chafer, A. Baz, M. Maurel, J. M. Blondy, E. Hugonnot, F. Scol, L. Vincetti, F. Gérôme, and F. Benabid, “Ultralow transmission loss in inhibited-coupling guiding hollow fibers,” Optica 4(2), 209–217 (2017).
[Crossref]

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

M. Zeisberger and M. A. Schmidt, “Analytic model for the complex effective index of the leaky modes of tubetype anti-resonant hollow core fibers,” Sci. Rep. 7(1), 11761 (2017).
[Crossref]

I. Del Villar, F. J. Arregui, C. R. Zamarreño, J. M. Corres, C. Bariain, J. Goicoechea, C. Elosua, M. Hernaez, P. J. Rivero, A. B. Socorro, A. Urrutia, P. Sanchez, P. Zubiate, D. Lopez, N. De Acha, J. Ascorbe, and I. R. Matias, “Optical sensors based on lossy-mode resonances,” Sens. Actuators, B 240, 174–185 (2017).
[Crossref]

2016 (3)

2014 (1)

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
[Crossref]

2013 (3)

2012 (1)

2011 (2)

2010 (3)

2009 (1)

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. H. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45(9), 1192–1200 (2009).
[Crossref]

2007 (1)

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318(5853), 1118–1121 (2007).
[Crossref]

2006 (1)

2005 (4)

2003 (1)

2002 (1)

1999 (1)

V. I. Neves and A. S. C. Fernandes, “Modal characteristics for W-type and M-type dielectric profile fibres,” Micro. & Opt. Tech. Lett. 22(6), 398–405 (1999).
[Crossref]

1997 (1)

1993 (1)

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993).
[Crossref]

1986 (1)

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Abdolvand, A.

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
[Crossref]

Abeeluck, A. K.

Aghaie, K. Z.

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. H. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45(9), 1192–1200 (2009).
[Crossref]

Ahmed, G.

Aleshkina, S. S.

S. S. Aleshkina, M. V. Yashkov, M. M. Bubnov, A. N. Guryanov, and M. E. Likhachev, “Asymptotically single-mode hybrid fiber for dispersion management near 1 µm,” IEEE J. Sel. Topics Quantum Electron 24(3), 1–8 (2018).
[Crossref]

S. S. Aleshkina, M. E. Likhachev, A. K. Senatorov, M. M. Bubnov, M. Y. Salaganskii, and A. N. Guryanov, “Low-loss hybrid fiber with zero dispersion wavelength shifted to 1 µm,” Opt. express 21(20), 23838–23843 (2013).
[Crossref]

Amsanpally, A.

Ando, R. F.

Archambault, J. L.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993).
[Crossref]

Arregui, F. J.

I. Del Villar, F. J. Arregui, C. R. Zamarreño, J. M. Corres, C. Bariain, J. Goicoechea, C. Elosua, M. Hernaez, P. J. Rivero, A. B. Socorro, A. Urrutia, P. Sanchez, P. Zubiate, D. Lopez, N. De Acha, J. Ascorbe, and I. R. Matias, “Optical sensors based on lossy-mode resonances,” Sens. Actuators, B 240, 174–185 (2017).
[Crossref]

I. Del Villar, I. R. Matías, F. J. Arregui, and P. Lalanne, “Optimization of sensitivity in Long Period Fiber Gratings with overlay deposition,” Opt. Express 13(1), 56–69 (2005).
[Crossref]

Ascorbe, J.

I. Del Villar, F. J. Arregui, C. R. Zamarreño, J. M. Corres, C. Bariain, J. Goicoechea, C. Elosua, M. Hernaez, P. J. Rivero, A. B. Socorro, A. Urrutia, P. Sanchez, P. Zubiate, D. Lopez, N. De Acha, J. Ascorbe, and I. R. Matias, “Optical sensors based on lossy-mode resonances,” Sens. Actuators, B 240, 174–185 (2017).
[Crossref]

Astapovich, M. S.

Bache, M.

Bang, O.

D. Jain, C. Markos, T. M. Benson, A. B. Seddon, and O. Bang, “Exploiting dispersion of higher-order-modes using M-type fiber for application in mid-infrared supercontinuum generation,” Sci. Rep. 9(1), 8536 (2019).
[Crossref]

Bariain, C.

I. Del Villar, F. J. Arregui, C. R. Zamarreño, J. M. Corres, C. Bariain, J. Goicoechea, C. Elosua, M. Hernaez, P. J. Rivero, A. B. Socorro, A. Urrutia, P. Sanchez, P. Zubiate, D. Lopez, N. De Acha, J. Ascorbe, and I. R. Matias, “Optical sensors based on lossy-mode resonances,” Sens. Actuators, B 240, 174–185 (2017).
[Crossref]

Baz, A.

Belanov, A. S.

A. S. Belanov and S. V. Tsvetkov, “High-index-ring three-layer fibres for mode-locked sub-1.3 µm fibre lasers,” Quantum Electron. 40(2), 160–162 (2010).
[Crossref]

Benabid, F.

B. Debord, A. Amsanpally, M. Chafer, A. Baz, M. Maurel, J. M. Blondy, E. Hugonnot, F. Scol, L. Vincetti, F. Gérôme, and F. Benabid, “Ultralow transmission loss in inhibited-coupling guiding hollow fibers,” Optica 4(2), 209–217 (2017).
[Crossref]

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318(5853), 1118–1121 (2007).
[Crossref]

Benson, T. M.

D. Jain, C. Markos, T. M. Benson, A. B. Seddon, and O. Bang, “Exploiting dispersion of higher-order-modes using M-type fiber for application in mid-infrared supercontinuum generation,” Sci. Rep. 9(1), 8536 (2019).
[Crossref]

Bierlich, J.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

Bigot, L.

Biriukov, A. S.

Black, R. J.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993).
[Crossref]

Blondy, J. M.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999).

Bouwmans, G.

Bradley, T. D.

Bubnov, M. M.

S. S. Aleshkina, M. V. Yashkov, M. M. Bubnov, A. N. Guryanov, and M. E. Likhachev, “Asymptotically single-mode hybrid fiber for dispersion management near 1 µm,” IEEE J. Sel. Topics Quantum Electron 24(3), 1–8 (2018).
[Crossref]

S. S. Aleshkina, M. E. Likhachev, A. K. Senatorov, M. M. Bubnov, M. Y. Salaganskii, and A. N. Guryanov, “Low-loss hybrid fiber with zero dispersion wavelength shifted to 1 µm,” Opt. express 21(20), 23838–23843 (2013).
[Crossref]

Bures, J.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993).
[Crossref]

Calvet, P.

Campopiano, S.

Chafer, M.

Chang, H.-C.

Chang, W.

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
[Crossref]

Chemnitz, M.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

Chen, Y.

Churbanov, M. F.

Contessa, L.

Corres, J. M.

I. Del Villar, F. J. Arregui, C. R. Zamarreño, J. M. Corres, C. Bariain, J. Goicoechea, C. Elosua, M. Hernaez, P. J. Rivero, A. B. Socorro, A. Urrutia, P. Sanchez, P. Zubiate, D. Lopez, N. De Acha, J. Ascorbe, and I. R. Matias, “Optical sensors based on lossy-mode resonances,” Sens. Actuators, B 240, 174–185 (2017).
[Crossref]

Coulombier, Q.

Couny, F.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multioctave optical-frequency combs,” Science 318(5853), 1118–1121 (2007).
[Crossref]

Cui, L.

Cusano, A.

Cutolo, A.

Dangui, V.

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. H. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45(9), 1192–1200 (2009).
[Crossref]

De Acha, N.

I. Del Villar, F. J. Arregui, C. R. Zamarreño, J. M. Corres, C. Bariain, J. Goicoechea, C. Elosua, M. Hernaez, P. J. Rivero, A. B. Socorro, A. Urrutia, P. Sanchez, P. Zubiate, D. Lopez, N. De Acha, J. Ascorbe, and I. R. Matias, “Optical sensors based on lossy-mode resonances,” Sens. Actuators, B 240, 174–185 (2017).
[Crossref]

de Sterke, C. M.

Debord, B.

Del Villar, I.

I. Del Villar, F. J. Arregui, C. R. Zamarreño, J. M. Corres, C. Bariain, J. Goicoechea, C. Elosua, M. Hernaez, P. J. Rivero, A. B. Socorro, A. Urrutia, P. Sanchez, P. Zubiate, D. Lopez, N. De Acha, J. Ascorbe, and I. R. Matias, “Optical sensors based on lossy-mode resonances,” Sens. Actuators, B 240, 174–185 (2017).
[Crossref]

I. Del Villar, I. R. Matías, F. J. Arregui, and P. Lalanne, “Optimization of sensitivity in Long Period Fiber Gratings with overlay deposition,” Opt. Express 13(1), 56–69 (2005).
[Crossref]

Delplace, K.

Dianov, E. M.

Digonnet, M. J. F.

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. H. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45(9), 1192–1200 (2009).
[Crossref]

M. J. F. Digonnet, H. K. Kim, G. S. Kino, and S. Fan, “Understanding air-core photonic-bandgap fibers: analogy to conventional fibers,” J. Lightwave Technol. 23(12), 4169–4177 (2005).
[Crossref]

Ding, W.

S. F. Gao, Y. Y. Wang, W. Ding, D. L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9(1), 1–6 (2018).
[Crossref]

Douay, M.

Duguay, M. A.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49(1), 13–15 (1986).
[Crossref]

Dunn, S. C.

Edavalath, N. N.

Eggleton, B. J.

Elosua, C.

I. Del Villar, F. J. Arregui, C. R. Zamarreño, J. M. Corres, C. Bariain, J. Goicoechea, C. Elosua, M. Hernaez, P. J. Rivero, A. B. Socorro, A. Urrutia, P. Sanchez, P. Zubiate, D. Lopez, N. De Acha, J. Ascorbe, and I. R. Matias, “Optical sensors based on lossy-mode resonances,” Sens. Actuators, B 240, 174–185 (2017).
[Crossref]

Erdogan, T.

Fan, S.

Fan, S. H.

K. Z. Aghaie, V. Dangui, M. J. F. Digonnet, S. H. Fan, and G. S. Kino, “Classification of the core modes of hollow-core photonic-bandgap fibers,” IEEE J. Quantum Electron. 45(9), 1192–1200 (2009).
[Crossref]

Farrell, G.

Fernandes, A. S. C.

V. I. Neves and A. S. C. Fernandes, “Modal characteristics for W-type and M-type dielectric profile fibres,” Micro. & Opt. Tech. Lett. 22(6), 398–405 (1999).
[Crossref]

Fokoua, E. N.

Frosz, M. H.

Gao, S. F.

S. F. Gao, Y. Y. Wang, W. Ding, D. L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9(1), 1–6 (2018).
[Crossref]

Gérôme, F.

Giordano, M.

Goicoechea, J.

I. Del Villar, F. J. Arregui, C. R. Zamarreño, J. M. Corres, C. Bariain, J. Goicoechea, C. Elosua, M. Hernaez, P. J. Rivero, A. B. Socorro, A. Urrutia, P. Sanchez, P. Zubiate, D. Lopez, N. De Acha, J. Ascorbe, and I. R. Matias, “Optical sensors based on lossy-mode resonances,” Sens. Actuators, B 240, 174–185 (2017).
[Crossref]

Gray, D. R.

Grigorova, T.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

Gu, S.

S. F. Gao, Y. Y. Wang, W. Ding, D. L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9(1), 1–6 (2018).
[Crossref]

Günendi, M. C.

Guryanov, A. N.

S. S. Aleshkina, M. V. Yashkov, M. M. Bubnov, A. N. Guryanov, and M. E. Likhachev, “Asymptotically single-mode hybrid fiber for dispersion management near 1 µm,” IEEE J. Sel. Topics Quantum Electron 24(3), 1–8 (2018).
[Crossref]

S. S. Aleshkina, M. E. Likhachev, A. K. Senatorov, M. M. Bubnov, M. Y. Salaganskii, and A. N. Guryanov, “Low-loss hybrid fiber with zero dispersion wavelength shifted to 1 µm,” Opt. express 21(20), 23838–23843 (2013).
[Crossref]

Habib, M. S.

Han, W.

D. Liu, Q. Wu, C. Mei, J. Yuan, X. Xin, A. K. Mallik, F. Wei, W. Han, R. Kumar, C. Yu, S. Wan, X. He, B. Liu, G.-D. Peng, Y. Semenova, and G. Farrell, “Hollow Core Fiber Based Interferometer for High Temperature (1000 °C) Measurement,” J. Lightwave Technol. 36(9), 1583–1590 (2018).
[Crossref]

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IEEE J. Quantum Electron. (1)

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Figures (8)

Fig. 1.
Fig. 1. A depressed-core optical fiber and its cross section and refractive index profile.
Fig. 2.
Fig. 2. (a) and (b) show the ray trajectory of a mode with n1 < neff = n2sinθ1 < n2 in a DCF and an equivalent asymmetric planar waveguide, respectively. (c) and (d) show the ray trajectory of a mode with n3 < neff = n1sinθ1 < n1 in the DCF and the equivalent asymmetric planar waveguide, respectively.
Fig. 3.
Fig. 3. Dispersion curves (neff vs. λ) of TE modes with neff corresponding to (a) n1 < neff < n2, (b) n1-0.001 < neff < n1+0.001, n1 = 1.445, n2 = 1.51. (c) shows the modal intensity and electric field vector distributions of TE modes whose positions (neff, λ) are indicated by the black circle dots in (a) and (b). The red vertical dashed lines in (a) and (b) indicate the resonant bands.
Fig. 4.
Fig. 4. The left panel (a)-(e) shows the normalized intensity distributions along the r-coordinate for the modes shown at points C-G in Fig. 3. The right panel (a′)-(e′) shows the zoomed-in part of (a)-(e), delineated by the vertical dashed lines.
Fig. 5.
Fig. 5. Dispersion curves (neff vs. λ) of TM modes with neff corresponding to (a) n1 < neff < n2, (b) n1-0.001 < neff < n1+0.001, n1 = 1.445, n2 = 1.51. (c) shows the modal intensity and electric field vector distributions of TM modes whose positions (neff, λ) are indicated by the black squares in (a) and (b). The black vertical dashed lines in (a) and (b) indicate the resonant bands.
Fig. 6.
Fig. 6. Dispersion curves (neff vs. λ) of HE1,N (red) and EH1,N (black) modes with neff corresponding to (n1-0.001 < neff < n1+0.001, n1 = 1.445. The black and red vertical dashed lines indicate the resonant bands.
Fig. 7.
Fig. 7. Dispersion curves (neff vs. λ) of HE modes with neff corresponding to (a) n1-0.001 < neff < n1+0.001, n1 = 1.445. (b) partially enlarged image of (a), indicated by the red frame. (c) shows the modal intensity and electric field vector distributions of HE modes whose positions (neff, λ) are indicated by the circles in (b). The black and red vertical dashed lines in (a) and (b) indicate the resonant bands.
Fig. 8.
Fig. 8. (a) Dispersion curves (neff vs. λ) of a depressed core fiber. (b) partially enlarged image of (a), indicated by the black dashed frame. The text labels and the corresponding dispersion curves are of the same color. The orange solid lines in (a) and (b) are for LPm,n modes in a conventional step-index fiber (nco = 1.445, ncl = 1, rco = 62.5 um and rcl = ∞). The black and red vertical dashed lines indicate the resonant bands.

Equations (21)

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λ N , c = 2 d n 2 2 n 1 2 [ N + 1 π tan 1 ( κ n 1 2 n 3 2 n 2 2 n 1 2 ) ] , κ = { 1 , for T E N modes n 2 2 n 3 2 for T M N modes
J ^ ( K p m + r m α 2 U 2 ) = 1 U 2 ( K ^ q m + s m α 2 U 2 ) , m = 0
J ^ ( K ^ p m + s 23 r m α 2 U 2 ) = s 21 U 2 ( K ^ q m + s 23 s m α 2 U 2 ) , m = 0
p m 2 + 2 ( 2 π α 2 U 2 2 ) 2 ( n 2 2 n 1 n 3 ) x 1 x 2 + x 1 2 x 2 2 [ J ^ ( K ^ p m + r m α 2 U 2 ) 1 U 2 ( K ^ q m + s m α 2 U 2 ) ] × [ J ^ ( K ^ p m + s 23 r m α 2 U 2 ) s 21 U 2 ( K ^ q m + s 23 s m α 2 U 2 ) ] = x 1 2 ( J ^ p m q m U 2 ) ( J ^ p m s 21 q m U 2 ) + x 2 2 ( K ^ p m + r m α 2 U 2 ) ( K ^ p m + s 23 r m α 2 U 2 )
α 2 = r 2 r 1 ,
u 1 = k 0 2 n 1 2 β 2 ,
u 2 = k 0 2 n 2 2 β 2 ,
ω 3 = β 2 k 0 2 n 3 2 ,
U 1 = u 1 r 1 , U 2 = u 2 r 1 , W 3 = ω 3 r 2 ,
J ^ = J m ( U 1 ) U 1 J m ( U 1 ) ,
K ^ = K m ( W 3 ) W 3 K m ( W 3 ) ,
p m = J m ( u 2 r 2 ) Y m ( u 2 r 1 ) J m ( u 2 r 1 ) Y m ( u 2 r 2 ) ,
q m = J m ( u 2 r 2 ) Y m ( u 2 r 1 ) J m ( u 2 r 1 ) Y m ( u 2 r 2 ) ,
r m = J m ( u 2 r 2 ) Y m ( u 2 r 1 ) J m ( u 2 r 1 ) Y m ( u 2 r 2 ) ,
s n = J m ( u 2 r 2 ) Y m ( u 2 r 1 ) J m ( u 2 r 1 ) Y m ( u 2 r 2 ) ,
s 23 = n 2 2 n 3 2 , s 21 = n 2 2 n 1 2 ,
V 12 2 = k 0 2 r 1 2 ( n 1 2 n 2 2 ) , V 23 2 = k 0 2 r 2 2 ( n 2 2 n 3 2 ) ,
x 1 2 = n 1 2 U 1 4 U 2 4 σ 0 2 V 12 4 , x 2 2 = n 3 2 α 2 4 U 2 4 W 3 4 σ 0 2 V 23 4 ,
σ 0 2 = ( β m k 0 ) 2 .
u 1 = β 2 k 0 2 n 1 2
J ^ = I m ( U 1 ) U 1 I m ( U 1 ) ,

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