The development of fiber laser sources and fiber optical components in the near- and mid-infrared (mid-IR) ranges is an important and topical problem of modern photonics. Fiber laser systems and nonlinear optical converters are in demand for a wide variety of applied and fundamental problems, including telecommunications, noninvasive medical diagnostics, laser surgery, sensing, spectroscopy, etc. [1
When designing nonlinear optical fiber devices, significant attention is paid to the control of group velocity dispersion, which is often of great importance in the development of laser sources based on supercontinuum generation, Raman solitons, and long-wavelength dispersive waves [1
]. Microstructuring is widely used for the development of fibers with required dispersion. Ordered structures with characteristic sizes of microns or submicrons are created using air holes or glass rods [15
]. Waveguide modes in microstructured fibers are formed as a result of the interference of waves arising from reflection and refraction at the refractive-index microinhomogeneities [18
]. In addition to the widespread photonic crystal fibers, in which several regular layers of air holes or glass rods are located around the core [16
], there are suspended-core fibers, in which a thin core is surrounded by one row of air holes with thin walls between them [14
]. In addition, the manufacture of solid microstructured fibers from low-temperature chalcogenide and tellurite glasses simultaneously was reported [36
Microstructuring can also be used to obtain multicore fibers, which is an emerging technology in fiber photonics. There exist structures formed by several regularly distributed cores with characteristic diameters of a few microns, in which the modes of individual cores overlap [37
]. In this case, from the solution of the Helmholtz equation, it is possible to find modes of the entire structure, which are called supermodes [38
]. For the fundamental supermode with the highest effective refractive index, the fields in each core are in-phase. For multicore fibers with cores arranged in a ring, the amplitude distribution for the in-phase supermode is the same in each core due to symmetry. As a rule, in the study of multicore fibers, primary attention is paid to the in-phase supermode [37
]. However, in the recent analytical, numerical, and experimental works, it has been shown that an out-of-phase supermode (in which the phases of the fields in neighboring cores differ by π) can also be of great interest due to its stability in different nonlinear regimes [41
]. Multicore fibers allow operation with pulse energies and powers higher than those of one-core fibers [41
] (however, for ultrashort pulses, spatial walk-off of supermodes should be taken into account [45
]). In addition, the supermode dispersion can vary significantly compared to the dispersion for a one-core fiber. For example, for nano-sized photonic wires, it has been shown theoretically that the dispersion of waveguide arrays can vary dramatically between different supermodes, and anomalous dispersion can be attained in coupled waveguides while a single waveguide has normal dispersion [47
], which was also confirmed experimentally [48
An important issue is the method of supermode excitation. To excite an in-phase supermode in a silica multicore fiber, a single-core fiber can be spliced with a multicore fiber after matching their claddings [49
]. More complicated fiber coupling devices, such as fan-in/fan-out fiber couplers, allowing access to one or more cores, can also be used [50
]. An out-of-phase supermode can be excited with free-space objective lenses [48
] or spatial light modulators [43
For the fabrication of microstructured multicore fibers operating in the near-IR range (as well as for the fabrication of any other types of near-IR fibers), silica glasses are the most popular due to the existing advanced technologies [37
]. The study of multicore fibers in the mid-IR is practically in its infancy, one of the reasons being the impossibility of using silica glasses due to high losses at wavelengths >2.3 μm. However, the development of technologies in the mid-IR range stimulates the search for new amorphous materials that combine a wide range of transparency, suitable nonlinear optical and/or laser characteristics, and optimal chemical and mechanical properties. Soft glasses, such as fluoride, tellurite, and chalcogenide, with low glass transition temperatures (~200–400 °C) are good candidates for mid-IR fiber development [26
]. Tellurite glasses based on tellurium dioxide TeO2
are transparent in the near- and mid-IR ranges (up to ~5–6 μm), have high chemical stability, and a high nonlinear refractive index [26
]; therefore, they are of great interest for the design of optical elements and nonlinear optical waveguide structures, including microstructured multicore fibers. To date, several works report the manufacture and study of multicore tellurite glass fibers [51
]. A numerical study of the dispersion of the fundamental supermode was carried out in [53
]. However, we are not aware of the study of the dispersion and nonlinear properties of an out-of-phase supermode in multicore fibers based on tellurite glasses.
In this work, we present a comprehensive numerical analysis of the dispersion and nonlinear properties of microstructured multicore tellurite glass fibers for in-phase and out-of-phase supermodes and compare them with the results for one-core fibers in the near- and mid-IR ranges. The out-of-phase supermodes in multicore tellurite fibers are studied for the first time, to the best of our knowledge.
4. Discussion and Conclusions
In this work, we calculated the dispersion and nonlinear characteristics of tellurite glass microstructured multicore fibers in the near- and mid-IR using the full-vector finite-element method. The choice of glasses for cores and cladding was based on the compatibility of their physicochemical properties suitable for manufacturing fibers [56
]. The numerical analysis presented in this work showed that the dispersion of multicore fibers could be effectively controlled by fitting the following parameters: core diameters, distance between centers, number of cores, as well as by choosing the operating supermode (in-phase or out-of-phase). Note that in the previous works devoted to multicore tellurite glass fibers, the out-of-phase supermode was not investigated, but numerical studies of dispersion of in-phase supermode have been reported [53
The fields propagating in individual cores overlapped and interacted, thus making a contribution to the dispersion of supermodes. Coupling coefficients were larger at longer wavelengths due to larger mode field areas. The closer the cores, the larger the coupling coefficient. So, dispersion as a function of wavelength for microstructured multicore fibers can differ significantly compared to the one-core fibers. Effective mode field areas and nonlinear Kerr coefficients of N-core fibers did not dramatically differ from the same characteristics of N independent one-core fibers (the corresponding values differ by no more than 1.5 times). The differences in the curve behavior were more noticeable for longer wavelengths.
For fibers with similar refractive indices of cores and cladding glasses (in this work, dn was 0.3%), the dispersion of individual one-core fibers did not differ much from the material dispersion. The microstructured multicore geometry made the following contribution: a single ZDW shifted toward longer wavelengths for in-phase supermodes and toward shorter wavelengths for out-of-phase supermodes compared to one-core fibers. The value of this ZDW shift was several tens of nm (up to ~100 nm).
With the use of fibers made of glasses with strongly different refractive indices of cores and cladding (in this work dn was almost 20%) with sufficiently thin cores of 2–3 μm, it was possible to achieve different qualitative and quantitative values of the group velocity dispersion in the near—and mid-IR ranges. For one-core fibers with such core properties, the dispersion already contained two ZDWs: The first ZDW was <2 μm and the second ZDW was >2.5 μm. For in-phase supermodes, the wavelength difference between the ZDWs decreased: The short-wavelength ZDW shifted to a longer wavelength, and the long-wavelength ZDW shifted to a shorter wavelength. For thin, closely spaced cores, there may arise a situation when ZDWs disappear, and the dispersion becomes all-normal. For out-of-phase supermodes, the wavelength difference between the ZDWs increased: the short-wavelength ZDW shifted to a shorter wavelength, and the long-wavelength ZDW shifted to a longer wavelength (or completely vanished). For out-of-phase supermodes, anomalous dispersion can be attained even at 1.55 μm, where standard Er-doped fiber lasers operate. This may be useful for the development of mid-IR sources utilizing a standard near-IR pump for supercontinuum generation, soliton self-frequency shift, etc.
The results of this work can be useful both for the design and development of dispersion-controlled microstructured multicore tellurite fibers and for the development of fibers based on other glasses. The presented qualitative conclusions about the behavior of dispersion curves for in-phase and out-of-phase supermodes can be used as a basis for choosing parameters of other pairs of glasses to obtain the desired dispersion characteristics.