1. Introduction

HC MOFs are a new form of optical fiber waveguides with unique properties which can be used for a variety of applications such as high power and ultra short pulse delivery, light – gas interactions and terahertz applications [1–5]. Such fibers confine electromagnetic fields inside a hollow core surrounded by a microstructured cladding. Light guided in such a way propagates primarily through an air and therefore has substantially lower absorption losses. For the first time, the confinement of light within a hollow core in silica HC MOF was demonstrated in [6]. To confine light in a hollow core one can use two main types of the fibers. The first type guide light via a photonic band gap mechanism under which the cladding doesn’t support modes for a certain range of wavelengths and propagation constants. Light in the core in those ranges is not able to couple with cladding modes and guides in the core with low losses. A well known example of such a fiber is silica hollow core photonic crystal fibers (HC PCFs) [7]. They have a hexagonal arrangement of holes in the cladding and their cores are formed by omitting several unit cells of the cladding [8]. The core boundary is antiresonant with the core modes to increase their confinement [9]. There is an example of silica HC PCF which demonstrated a low loss transmission window in the mid – IR [10]. Another well known example of this HC MOF type is an omniguide fiber, the cladding of which is a Bragg reflector made of soft glasses and polymers [11] or polymer ring structured Bragg fibers where the cladding has concentric rings of holes [12]. Guiding with a loss below the material loss was demonstrated in omniguide fibers for the first time. The second type of HC MOFs doesn’t support photonic band gaps [7] and its core modes have a weak coupling with cladding modes. This is the so called “low density of state guidance” [13]. For example, this type of HC MOFs has kagome lattice claddings [14]. It has a relatively higher transmission loss in comparison with bandgap HC PCFs but with a much larger bandwidth. Recently, there has been much interest in other broad band HC MOFs of the second type in a terahertz spectral region with a cladding formed by a periodic arrangement of tubes in a triangular lattice [4,15,16]. Such HC MOFs were called tube lattice fibers (TLFs). TLFs demonstrate very interesting properties in a terahertz spectral region such as a transmission band width of several hundred of GHz, low loss and low dispersion. In theoretical work [16] the TLFs waveguide mechanism was analyzed and it was demonstrated that waveguide properties of a single cladding tube have strong impact on the fiber waveguide mechanism. Some our results that confirm the conclusions made in [16] will be reported in Section 2 of this paper.

In our paper we demonstrated numerically and experimentally that simplified HC MOFs similar to TLFs can be used for guidance with a low loss in all spectral regions beginning with visible light up to the mid infrared. Moreover, it is not necessary to use HC MOFs composed of many tubes to achieve a low loss in a multimode waveguide regime. For the first time, HC MOFs with simplified cladding consisting of six thin bridges suspending the core surround were demonstrated in [17,18]. However, a curvature of the core boundary was equal to zero (flat surfaces) for these fibers. HC MOFs proposed in our paper have a cladding consisting of only eight silica capillaries and they can give a waveguide transmission in the spectral region > 3.5 μm despite a high material loss of silica in this spectral region [19]. In our opinion, such a waveguide regime is achieved due to a negative curvature of the core boundary and individual scattering characteristics of each element of the cladding. Moreover, the size of the core diameter with respect to the used light wavelength also plays an important role in establishing a spectral range for such a broad band transmission. Here, it is necessary to clarify what exactly is meant by a negative curvature of the core boundary. Further it will be assumed that if the surface normal to the core boundary is co directional with a radial unit vector in a cylindrical coordinate system then we have a positive curvature of the core boundary and vice versa. To achieve the lowest total loss it is necessary to minimize an axial flow inside the walls of the capillaries. In other words, it is necessary to maximally restrict the intensity of the core mode interaction with the walls of capillaries in space. In our case, a negative curvature of the core boundary is yielded by a cladding consisting of one row of capillaries and by changing in their number. This factor gives a possibility of decreasing in total loss dramatically in all low loss spectral regions including long wavelength bands. The first realization of HC MOF [6] had a negative curvature of the core boundary. However, the symmetry of the capillary arrangement in the first row of the cladding forming the core boundary was different from the one used in our work. For the first time, an effect of decreasing in the loss level due to a negative curvature of the core boundary was observed for kagome lattice HC MOF [20]. Our results show that this phenomenon can be used for managing a location of transmission bands in all spectral regions beginning with the UV up to the mid infrared depending on HC MOFs geometry sizes and composition of capillary glass.

The paper is organized as follows. In Section 2, we consider an example of HC MOFs with a negative curvature of the core boundary and carry out some numerical analyses of its waveguide properties. In Section 3, the waveguide transmission in all bands from the near to mid infrared is experimentally demonstrated for HC MOFs with a cladding consisting of eight capillaries. Section 4 contains the conclusions.

2. Numerical demonstration of broad band transmission of HC MOF with negative curvature of the core boundary

In this section we consider an example of HC MOFs with a negative curvature of the core boundary and compare its transmission characteristics to those of a simple tube. Let us consider two types of HC MOFs consisting only of one row of eight capillaries and rods in the cladding (Fig. 0.1). It is assumed that both HC MOFs are made of silica. The dependence of real and imaginary parts of the refractive index in the near and mid infrared is taken according to [19]. To analyze numerically the waveguide properties of these multimode HC MOFs we calculated a total loss for a fundamental mode (FM) (Fig. 2 ) depending on the wavelength from a near to mid infrared spectral region for two sets of geometry sizes. We used the Femlab 3.1 commercial packet based on the finite element method for the calculation of the FM total loss.

 

Fig. 2 Fundamental mode of HC MOF with the cladding consisting of capillaries.

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The first considered HC MOFs have an effective air core diameter D_core = 36 μm, where the effective air core diameter is regarded as a minimum distance between two opposite capillaries – rods of the cladding (Fig. 1 (left)). The outer diameter of capillary and rod is d_out = 22.5 μm and the inner diameter of capillary is d_ins = 0.76*d_out. Moreover, we calculated waveguide loss for the FM in the case of a simple dielectric tube made of silica with the same value of D_core using analytical expression derived in [21]. The results are shown in Fig. 3(a) . As expected, the waveguide losses of the tube are several orders higher than those of the HC MOFs and the tube has no waveguide regime at such values of λ/D_core [21], where λ is a wavelength. The total loss of the HC MOFs with the cladding consisting of rods (Fig. 1 (right)) is much lower than that of the tube, though the waveguide regime is not effective in the considered wavelength range.

 

Fig. 1 HC MOF with a negative curvature of the core with cladding consisting of one row of capillaries (left) and with the cladding consisting of one row of rods (right).

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Fig. 3 (a) Total loss for HC MOFs with D_core = 36 μm in the case of cladding consisting of capillaries (square), rods (circle) and simple dielectric tube (triangular) (b) the total loss for HC MOFs with D_core = 68 μm and all notations is the same as in (a).

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Its ineffectiveness in this spectral range can be explained by poor scattering properties of the rods at such values of ratio λ/d_out. The single rod of the cladding has a great number of its own resonance wavelengths and, as a consequence, a high density of states of leaky modes with a low quality factor. The resonances of these own states overlap with each other in all spectral ranges and there is no possibility of an emergence of transmission bands with a low loss. In turn, a decrease in the total loss of the HC MOFs with respect to one of the tubes is connected partly to a change in a sign of the core boundary curvature which leads to a decrease in intensity of the core modes interaction with the rods material.

In order to obtain an effective waveguide regime in the considered wavelength range it is necessary to combine both factors. It is necessary to lower the density of states of a single element of the cladding and to keep the negative curvature of the core boundary. It can be achieved for HC MOFs with a cladding consisting of capillaries with the same value of d_out. The spectrum for capillary has a lower density of states and it is shifted with respect to one of the rods. This fact is analogous to a well known difference between spectrums of open and closed resonators. From Fig. 3(a) it can be seen that HC MOFs with such a cladding has several bands with low losses from λ = 1 to λ = 3 μm in which the waveguide regime is possible and rather effective.

In order to check this effect at other values of geometry parameters and to obtain the waveguide regime at longer wavelengths we increased the effective air core diameter up to D_core = 68 μm and the value of ratio d_ins/d_out = 0.8. The outer diameter of the capillary was d_out = 43.3 μm. The calculations results are shown in Fig. 3(b). The qualitative behavior of losses depending on the wavelength is the same as in the previous case. Quantitatively, one can observe no large decrease in the level of loss for HC MOFs with a cladding consisting of rods as well as for the tube. As for HC MOFs with a cladding consisting of capillaries one can observe a shift in all transmission regions to a longer wavelength and a decrease in the level of a total loss. We explain this fact not only by an increase in D_core but also by an increase in the d_ins/d_out ratio and, as a consequence, by a decrease in density of the capillary states. Since the locations of the bands with a high loss are determined by the thickness of the capillary walls (ARROW model [22]) these bands (resonances) can be broadened under thickness variations of the capillary walls. Evidently, this fact leads not only to a decrease in widths of the transmission bands but also to an increase in a total loss in these bands due to a partial overlap between the neighboring resonances. The same results were obtained in [16]. This assumption will be also confirmed in the next section.

In such a way, it is possible to obtain low loss transmission band for HC MOFs with a cladding consisting of one row of capillaries even in a mid infrared wavelength region (λ > 4 μm) despite of very high material losses of silica. In the next section we will demonstrate experimentally the low loss transmission from a near to mid infrared wavelength range.

3. Experimental results

To confirm the numerical results obtained in the previous section we have made HC MOFs with a cladding consisting of eight capillaries. The preform was made by the stack and draw method. The silica tube (Suprasil F-300) was drawn to capillaries with d_out = 4 mm and with d_ins = 3 mm. The eight capillaries were inserted into a silica tube with d_out = 20 mm and with d_ins = 15 mm to obtain the required structure. Then, the construction was fused by an oxygen – hydrogen burner and the process temperature being adjusted so as to splice the capillaries together as shown in Fig. 4 without noticable deformation. Thereafter, the preform was drawn at a standard drawing tower. During the drawing process excessive inner pressure was applied to the capillaries to prevent them from collapsing. An example of such HC MOFs with D_core ≈36 µm is shown in Fig. 4.

 

Fig. 4 The preform (left) and cross section of HC MOF (right) with the cladding consisting of eight capillaries and D_core ≈36 μm.

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We drew several samples with outer diameters equal to 80, 103, 125 and 206 µm and with values of ratio d_ins/d_out = 0.76. The results of measurements of transmission bands in the near infrared for the first three samples are shown in Fig. 5 .

 

Fig. 5 The measured transmission bands in a near infrared spectral region for HC MOFs with outer diameters of 80 μm (solid), 103 μm (dashed) and 125 μm (dotted).

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As it can be seen, the HC MOF with the outer diameter of 125 μm has the highest loss in this spectral region. It occurs due to the fact that the capillaries constituting the cladding of this HC MOF has the thickest walls and, as a consequence, the highest density of the capillary states in this spectral range. Moreover, as it can be seen from Fig. 6 , the structure of the core modes is different in the case of HC MOFs with the outer diameter of 80 μm and 125 μm, for example.

 

Fig. 6 (a) The core mode of HC MOF with the outer diameter of 80 µm and d_ins/d_out = 0.76 in one of the transmission band in near infrared taken by a CCD camera (b) the core mode of HC MOF with the outer diameter of 125 µm in the same spectral region.

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In the case of HC MOF with outer diameter of 80 μm the FM is observed in the considered transmission band just as in the case of HC MOF with the outer diameter of 125 μm the higher core mode is observed.

Then, HC MOF with the outer diameter of 206 μm was chosen to observe the waveguide regime at longer wavelengths. The total length of the measured HC MOF was equal to 63 cm. The effective core diameter of the HC MOF equals approximately to 68 μm. A Fourier spectrometer JFS – 113v was used as a light source to investigate the transmission of the HC MOF in a broad spectral region from 1 to 5 μm. The observed transmission bands for this HC MOF are shown in Fig. 7(a) . As it was expected, several transmission bands are observed from 1 up to 4 μm in which the waveguide regime is possible. The calculated bands for this HC MOF are shown in Fig. 7(b). As it can be seen, there is a very close correspondence between the edges of measured transmission bands and the calculated ones. The level of the total loss in all the calculated bands is relatively low and weakly dependent on the refractive index of the capillary walls material. Moreover, it can be seen from comparing Fig. 3(b) to Fig. 7(b) that it is possible to shift all the bands further to longer wavelengths in a mid infrared spectral region by changing the thickness of the capillary walls.

 

Fig. 7 (a) The measured transmission bands for HC MOF with the outer diameter of 206 µm and d_ins/d_out = 0.76 from a near to mid infrared spectral region (b) the calculated bands for the same HC MOF.

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This fact points to a strong correlation between the density of single capillary states and the total loss of the HC MOFs in long wavelength bands. In such a way, our assumptions about the waveguide mechanism of such HC MOFs with a simple construction of the cladding and the origin of their low loss were confirmed.

4. Conclusions

We have demonstrated numerically and experimentally an existence of several transmission bands with a total loss much lower than the material loss of silica in a mid infrared spectral region for simple hollow core HC MOFs with a cladding consisting of several capillaries. In our opinion, such HC MOFs can guide a radiation from a near to mid infrared spectral region due to the two following factors: the negative curvature of the core boundary and the low density of states of scattering elements of the cladding. Moreover, it is not necessary to create a complicated structure of a cladding with many rows of scattering elements because, according to our results, all main waveguide properties are determined by the first row of the cladding elements. It seems possible to create HC MOFs based on the same guiding principles of transmission in all spectral regions beginning with the UV. It is possible that these HC MOFs will be potential candidates for transmission of high power CO₂ laser radiation. Full analyses of HC MOFs with a negative curvature of the core boundary will be given in our further publications.