We have theoretically and experimentally investigated dual-core photonic bandgap fibers (PBGFs), which consist of a cladding with an array of high-index rods and two cores formed by omitting two nearby rods. We find novel features in their coupling characteristics such as maxima and minima in coupling length, complete decoupling of the cores, and an inversion of the usual ordering of supermodes so that the odd supermode has the higher propagation constant. This behavior is understood by considering the field distribution in the rods between the cores.
© 2007 Optical Society of America
Photonic bandgap fibers (PBGFs) confine light in a core by a photonic bandgap (PBG) of the photonic crystal cladding. All-solid PBGFs consist of a two-dimensional array of high-index rods in a lower-index background , commonly made of doped and undoped fused silica, with the core formed by omitting one of the rods [2, 3]. Since they can exhibit the unusual spectral and dispersive characteristics of all PBGFs while being easier to fabricate and splice to conventional fibers than hollow-core PBGFs, all-solid PBGFs can not only be used as a simple model for studying all kinds of bandgap fibers but also have potential applications in optical communication and sensor devices.
Multi-core photonic crystal fibers (PCFs) are attractive due to their design flexibility and the ease with which they can be fabricated by using the stack and draw procedure. Although index-guiding multi-core PCFs have been widely investigated experimentally and theoretically [4–6], no experimental results on multi-core PBGFs have been reported before now. However, theoretical studies have shown that such fibers can have some remarkable coupling properties. Skorobogatiy et al  showed that resonant coupling between two hollow cores can be induced by introducing defects between them, and Wang et al  showed that decoupling (zero power transfer between closely-spaced cores) can occur between hollow cores due to interactions with surface modes. Decoupling of parallel dielectric waveguides is possible if the index between them is a suitable function of the transverse coordinates , which has been widely investigated in 2-D photonic crystal slab waveguides [10–13]. For fibers like the all-solid PBGF, Laegsgaard  found that coupling between two cores is affected by the modes of the rods around them, such that the coupling length-versus-frequency curve can exhibit maxima if the two cores are separated far enough. However, no comment was made about the ordering of the even and odd supermodes, or the possibility that they might become degenerate and lead to complete decoupling.
In this paper we report the first experimental measurements on photonic bandgap fibers with multiple cores, specifically all-solid silica PBGFs with two closely-spaced cores. Numerical calculations in section 2 demonstrate that extrema in coupling length versus frequency (both maxima and minima), complete decoupling, and a reversal of the usual ordering of the supermodes are all possible in such fibers. In section 3 we describe the fabrication and experimental characterization of the fibers. Although differences between the modeled and actual index profiles in the high-index rods significantly affect coupling behavior, the experimental results also exhibit extrema in coupling length and complete decoupling, in good qualitative agreement with the numerical results. Finally, we give conclusions in Section 4.
2. Theory and modeling
2.1 Properties of conventional dual-core fibers
In a dual-core fiber with two identical single-mode cores, the modes of the individual cores interact via their evanescent fields and form a pair of supermodes, one even (symmetric) and one odd (antisymmetric). The propagation constants β of the two supermodes are slightly different from the modes of the individual cores and from each other. The shift in β is determined by the coupling strength between the cores and can be expressed as a coupling coefficient, defined by an overlap integral between the individual core modes over one of the cores .
When light is incident on just one core at the fiber endface, both supermodes are equally excited. The phase difference they acquire as they propagate effectively causes light to oscillate from one core to the other and back. The coupling length Lc is the distance along the fiber in which there is total transfer of power from one core to the other:
where β even and β odd are the propagation constants of the even and odd supermodes respectively, n even and n odd are their corresponding effective indices, k = 2π/λ and λ is the free-space wavelength. The coupling length Lc varies with the strength of coupling between the cores: Lc is short if coupling is strong, but becomes infinite if there is complete decoupling.
Conventional dual-core fibers are well-understood and are known to have two general properties. First, Lc increases monotonically with frequency and core separation as the overlap of the mode of one core with the other core decreases. Second, the effective index of the even supermode is always greater than that of the odd supermode. This is because the fundamental modes of the individual cores have positive phase everywhere so their overlap is always positive. These properties will be contrasted with those of PBGFs in the following sections.
2.2 Coupling in dual-core PBGFs
The above properties generally apply to dual-core index-guiding PCFs as well. However, if the cores are doped to have a depressed refractive index, the core modes can become less well-confined as frequency increases , giving a minimum in the frequency dependence of Lc . For the same reason, such a minimum can occur in dual-core PBGFs too . Furthermore, because the guided mode in the core of a PBGF is not the fundamental mode of the fiber (the true fundamental mode being a cladding state), there is no prohibition on the ordering of the even and odd supermodes being inverted in effective index.
Figure 1 shows the cores of the modeled PBGFs, chosen to represent actual structures investigated experimentally. The cladding is a triangular array of high-index rods of diameter d = 0.7Λ, where Λ is the rod pitch. The cores are formed by omitting two rods from the array. The refractive indices of the rods and low-index background are 1.47288 and 1.457 respectively, a contrast of ∼1%. Material dispersion has no impact on our results and is neglected. We studied two core arrangements with different separations Dc (2Λ and √3Λ, as shown) and symmetry axes. The x-axis passes through the cores and the y-axis lies on the line of reflection symmetry.
Figure 2 is the photonic density of states (DOS) of the cladding calculated by the fully-vectorial fixed-frequency plane-wave method . The edges of the bandgaps were also found using the MIT plane-wave package  and were consistent with Fig. 2.
In the 2nd and the 3rd bandgaps, the effective indices neff and field distributions of the even and odd supermodes of the dual-core PBGFs were calculated using the FEMLABTM full-vector finite element method package. The effective indices are shown in Fig. 3. Only the y-polarization is shown because the difference between x- and y-polarized supermodes is small (as expected for a weak index contrast ). The coupling length Lc obtained using Eq. (1) is plotted in Fig. 4 for both polarizations.
Some properties not seen in conventional dual-core fibers occur in Figs. 3 and 4. Firstly, for Dc = √3Λ, complete decoupling occurs near the blue (high-frequency) edges of both bandgaps: the dispersion curves of the even and odd modes intersect and Lc becomes infinite. Secondly, the usual ordering of even and odd modes is reversed (n odd > n even) in the third bandgap but not in the second, for both core separations except beyond the decoupling points for Dc = √3Λ. Thirdly, some extrema (maxima and minima) are seen in the Lc curves.
The core-guided mode has a presence in the high-index rods in all-solid PBGFs, so we expect the field distributions in the rods to play a key role in the coupling properties of dual-core PBGFs. The rods directly between the cores are particularly important because this is where even and odd supermodes are most clearly different. Since the supermodes can be built up from the modes of each core in isolation, we calculated the mode fields of a single-core PBGF with the same cladding structure. We chose frequencies near the bandgap edges, where the influence of the rods will be strongest . The fields are shown in Fig. 5. Also marked are the rods that could be omitted to form each dual-core fiber of Fig. 1, and the corresponding symmetry (y) axes. The field of the rod-guided mode from which the adjacent band is formed is shown in an inset to each plot for comparison. This represents the cladding mode that interacts most strongly with the core mode. As shown in Fig. 2, these modes are LP11 and LP21 in the 2nd bandgap, and LP02 and LP31 in the 3rd bandgap.
Firstly, we discuss the decoupling seen at the blue edge of both bandgaps for Dc = √3Λ. This can be explained by an avoided-crossing effect between one supermode and a cladding mode of matching symmetry at the band edge. If the dispersion curves of two modes of the same symmetry approach, the curves repel each other. As shown in Fig. 5(b) for the blue edge of the 2nd bandgap, the dominant field in the high-index rod is LP21, which is odd with respect to the symmetry axis for Dc = √3Λ and even for Dc = 2Λ. Thus an avoided crossing occurs for the odd supermode if Dc = √3Λ but for the even supermode if Dc = 2Λ, in each case pushing up its dispersion curve as the band edge is approached. For Dc = √3Λ this opposes and eventually overwhelms the n even > n odd ordering in the middle of the bandgap, causing the curves to cross and give a decoupling point, whereas for Dc = 2Λ it reinforces that ordering so the curves do not cross. The same happens at the blue edge of the 3rd bandgap: compared to the 2nd bandgap, the symmetries in the rods are opposite but the mode ordering in the middle of the bandgap is also opposite (n odd > n even).
At the red edges of the bandgaps, there are no strong symmetry differences between the two structures. In both cases the mode with the lower index is pushed down and no decoupling occurs.
Secondly, we consider why the supermodes can have the reverse n odd > n even ordering never seen in conventional dual-core fibers, even far from the band edges where the avoided-crossing effect described above is weak or absent. This is because the bandgap-guided mode of a single-core PBGF is not the fundamental mode of the fiber (the true fundamental mode being a cladding mode). Its field therefore changes sign from place to place. If the field has opposite signs in the core and in the site where the second core would be, the overlap integral  discussed in section 2.1 becomes negative and so does the effective index difference of the supermodes. (We note that the same can apply to higher-order modes in conventional dual-core fibers.)
In Fig. 6 we examine field distributions of a single-core PBGF along the line joining the core to the site where the second core would lie for the Dc = 2Λ case (chosen because decoupling due to avoided crossings is absent). The CUDOS software  was used for this calculation. In the 2nd bandgap the field is mainly positive over the second core whereas in the 3rd bandgap it is mainly negative, thus leading to the observed ordering of the supermodes in both cases. (It should be noted that the overlap integral should strictly be taken over the difference between the two-core and one-core index distributions , ie just the missing rod, where the dominant sign of the field in Fig. 6 is not so obvious.)
Thirdly, it is not easy to predict the extrema in the Lc curves of Fig. 4 intuitively. The varying strength of confinement across the bandgap , the often-conflicting influences of avoided crossings at both band edges and the unfamiliarity of the field distributions of even a single-core PBGF adds up to a complex picture that is best resolved by resorting to numerical calculations for each case.
We fabricated two types of dual core all-solid-PBGF with the core separation distance Dc of 2Λ and √3Λ. Our starting material was a germanium-doped silica rod with the refractive index profile shown in Fig. 7(a). This rod was overclad by a pure silica tube and drawn down to canes with diameters of ∼1 mm. The canes were stacked and then the cores were formed by replacing two of the doped canes with pure silica canes. We made the two different core designs in one stack to obtain otherwise identical fibers. The stack was jacketed and drawn down to canes with diameter of ∼ 6 mm, and then the canes were drawn down to form bandgap fibers such as shown in Figs. 7(b) and 7(c). The fiber was coated in a polymer coating during the drawing. We drew fibers with outer diameters about 210 μm in order that the second bandgap was located around the 1.55 μm wavelength band. The nominal ratio of the diameter of the raised-index rods to the pitch Λ was d/Λ = 0.75, and the pitch Λ was about 9.9 ± 0.1 μm in the final fiber.
We measured the spectral transmission through our fibers using a fiber-based supercontinuum source . We coupled the light into one of the two cores and measured the transmitted spectra for each of output cores using an optical spectrum analyzer. At the same time, the polarization dependence was also measured using a polarizer. The sample was kept straight to prevent undesirable coupling to the cladding modes. Four low-loss transmission windows were observed within the range of 500 – 1700 nm, which correspond to the 2nd to 5th bandgaps. A typical spectrum of the 2nd and 3rd bandgaps from a fiber with Dc = 2Λ is shown in Fig. 8(a). The complementary intensity spectra from the two cores (within the low-loss transmission windows) show the wavelength dependence of the coupling length Lc, and are very different from those of index guided PCFs .
In order to measure Lc directly we repeated the spectral transmission measurements while progressively cutting the fiber back. Sample lengths were varied in the range of 50 – 400 mm to see a suitable number of spectral peaks in the transmission window. The cutback lengths used were in the range 1 – 5 mm and this was always less then 2Lc. The measurement error of each cut length was less than 1 mm. The normalized intensity spectrum of each core [Fig. 8(b)] showed a sine curve and was shifted in phase with each cutback. When the phase change was amounted to 2π, the total length cut was 2Lc. The sample length L can be expressed L = 2nLc (n: natural number) at the intensity peaks of core 1 (input and output core are same), and L = (2n - 1)Lc at the intensity peaks of core 2 (input and output core are different). We identified peaks whose phase change was exactly 2π after the cutbacks, and determined the values for n and Lc at that wavelength. The n of adjacent peak was (n + 1) if the phase change is larger and (n - 1) for the opposite case, so the Lc of other peaks could be obtained at the same time.
Figure 9 shows the experimental and calculated results of the Lc. The calculation was done using the refractive index profile of Fig. 7(a), and d/Λ = 0.75. The calculated results of the Lc are quite different from that of the section 2 due to the different refractive index profile, but they still exhibit the minimum and decoupling points in which we are interested. The experimental and calculated Lc curves show good agreement in terms of showing the minimum, and the steep slope at some bandgap edges. The numerical values of the Lc agree well in all cases in spite of measurement errors in the sample length, the pitch and the refractive index profile. The decoupling point which exists (in the calculations) near the red edge of the 2nd bandgap of Dc = √3Λ fiber has not been directly observed in our experimental measurements because of the low transmission around the bandgap edge, but the steep Lc curve in this spectral region is an indication of its presence. Most of the measured Lc curves were slightly shifted towards higher kΛ, and this might be due to the change of index profile during the fiber fabrication process.
As expected, the weak index contrast in the fibers means that the differences between the coupling lengths for x- and y-polarized supermodes are small. The measured polarization splittings are comparable with the calculated values.
We have theoretically and experimentally demonstrated novel coupling properties in dual-core all-solid PBGFs. As well as the existence of extrema in coupling length versus frequency as described in Ref. , our numerical modeling shows for the first time that complete decoupling as well as the inversion of the familiar supermode ordering of n even > n odd can occur. These phenomena can be explained with reference to the field distribution in the high-index rods between the cores.
We have made the first dual-core photonic bandgap fibers and measured their properties. The experimental results also exhibit extrema in coupling length and complete decoupling, showing good qualitative agreement with the numerical results. These fibers already have potential for applications as optical communication devices, such as WDM interleavers and broadband directional couplers. We anticipate that this platform will lead to other designs with enhanced characteristics for different applications.
This work was supported by Sumitomo Electric Industries Ltd., the UK EPSRC, the National Basic Research Program of China (973 project, grant 2003CB314906), and the project of excellent postdoctoral training in Europe of Tianjin government, China. We thank Mr Alan George for help in fabricating the fibers, Greg Pearce and David Bird for making their fixed-frequency plane-wave code available, and David Bird for some early calculations of coupling lengths.
References and links
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