IR transmissive As-S glass and As-Se glass triangular photonic band gap fiber structures are theoretically modeled. The potential for propagation of air-guided modes in the defect regions of these fibers is demonstrated by the large out of plane two-dimensional photonic band gaps found in these structures. Fiber design for IR light propagation is discussed.
© 2003 Optical Society of America
Transmission of high average power and high peak power laser transmission through fibers is limited by material breakdown and nonlinearities due to interaction with the fiber material. Air guiding photonic band gap (PBG) fibers can potentially overcome these limitations and allow higher power laser transmission than can be realized in conventional solid core fiber. In these fibers, light is guided in a central air core defect region and confined by means of out of plane two-dimensional photonic band gaps formed by a periodic structure of air holes in the cladding region. Development of PBG fibers in silica glass has progressed rapidly. Currently, losses in silica based PBG fiber of 13 dB/km at 1.5 µm have been reported  and PBG fiber with low loss transmission windows in the near IR are available from several companies [2,3].
For signal and laser power transmission in the IR beyond the transmission window of silica glass (~2 µm), the applicability of silica based PBG fiber is questionable. In an ideal PBG structure with an infinite number of hole layers, the propagating modes can be contained entirely in the air defect region. Practically, however, the finite number of hole layers, variations in hole periodicity and deformation of the air hole size and shape results in penetration of the mode field into the “cladding.” This can result in significant loss for signals at wavelengths that are highly absorbed by the PBG glass matrix. Propagation of high power energy at these wavelengths can lead to heating and damage of the fiber. Thus, while PBG fiber structures can be designed to propagate light beyond 2 µm, the high absorption of silica in the multiphonon edge will result in significant loss.
To circumvent this problem, we have explored development of PBG fibers in IR transmissive chalcogenide glass. These glasses are composed of the chalcogen elements Se, S, or Te with the addition of other elements such as Ge, As, and Sb to form stable glasses. Solid core effective index guiding microstructured fibers have already been demonstrated in gallium lanthanum sulfide (GLS) glass . For PBG fiber modeling, As-S and As-Se glasses were chosen. The glasses have been shown to be chemically and mechanically durable, possess wide glass forming regions and can be fabricated into low loss fiber . Typical losses of fiber drawn in our laboratory are ~0.1 to 0.2 dB/m for As-S fiber and ~0.2–0.4 dB/m for As-Se fiber. The transmission window for fibers based upon As-S and As-Se glass is approximately 1–7 µm and 1.5–10 µm, respectively. Conventional core/clad fibers drawn from these glasses have demonstrated high power handling capability in the IR beyond 2 µm [6–8]. PBG fiber fabricated from these glasses should demonstrate significantly lower loss and higher power handling capability.
Chalcogenide glasses have much higher indices than silica glass. Indices of ~2.4 for As-S glass (composition As39S61) and ~2.8 for As-Se glass (composition As39Se61) are typical. As of yet, no modeling has been performed on these high index glasses. Modeling for silica-air PBG structures has demonstrated band gaps with air fills of 45% and 70% in triangular structures [9,10] while experimental structures have been demonstrated with air fills of approximately 89%. 
Several programs are available for modeling PBG structures. In this paper, we have utilized Rsoft’s BandSolve v.1.1.6 to evaluate the PBG fiber structures . The program utilizes a full-vectorial plane wave expansion method for calculating PBG structures. This method has been well described in the literature [10,13,14] and is based upon casting Maxwell’s equations in a periodic structure as an eigenvalue problem with the electromagnetic field expanded in a basis of plane waves.
Modeling was performed on As-S and As-Se PBG fiber with triangular air hole pattern and air fills of 20 to 90% in 5% increments. For the triangular air hole structure, the air fill is limited to a maximum value of 0.907 before the air holes touch. For both As-S and As-Se, the index at a wavelength of 4 µm was utilized for the calculations. Material dispersion data indicates that the index varies by <±0.005 over the atmospheric transmission window from 3–5 µm in these glasses. For modeling, the first 32 bands were calculated for each structure and gap maps showing the variation in band gap with propagation constant were generated. The propagation constant is normalized with respect to the center-to-center air hole distance, or pitch, Λ and is represented by βΛ.
Bound modes in the air defect region should lie near the air line at β/k=1. The larger the band gap width, ΔβΛ crossing the air line, the larger the number of modes that can be supported by the fiber for a given defect core size and operating wavelength. The larger the band gap height, ΔkΛ, the broader the spectral range for guidance of a given mode. Fig. 1(a) through 1(d) shows representative gap maps for As-S PBG fiber of 40%, 60%, 75%, and 90% fill, respectively. Four general trends in the band gaps crossing the air line are observed. In the first region, for air fills up to 40%, no air line band gaps appear. In the second region, for air fills of 45% to 65%, several closely spaced band gaps appear which cross the air line at βΛ=4.5 to 9.5. The air line crossings peak in number and width at air fill values of around 55% to 60%. The number and width of the band gap crossing decrease in the third region at air fills of 70% and 80% while in the fourth region, for fills of 85% to 90%, several large band gaps crossing the airline open up at βΛ>7.
For As-Se, the trend is similar to As-S as shown by the representative gap maps for 40%, 60%, 75% and 90% fill in Fig. 2(a) to 2(d). Again four general regions are observed. No band gaps cross the air line for air fills up to 40%. For air fills between 45 to 65%, several closely spaced band gaps cross the airline at βΛ=4.5 to ~8.5, peaking in number and width at air fills of 55% and 60%. As with As-S, the number and width of the band gaps crossing the airline decreases at fills of 70% and 80% while larger band gaps crossing the airline open up at high values of βΛ for fills of 85% and 90%.
As mentioned above, the larger the band gap width, ΔβΛ crossing the air line, the larger the number of modes that can be supported by the fiber for a given defect core size and operating wavelength. The larger the band gap height, ΔkΛ, the broader the spectral range for guidance of a given mode. Thus, to quantify the light-guiding ability of the structures, we define the gap area of the band gaps crossing the air line as the product of these two values. In Fig. 3, the gap area of the largest band gap crossing the air line is plotted vs. air fill for both As-S and As-Se PBG fiber. We note that the trends observed above are mirrored in these graphs. No band gaps crossing the air line exist for air fills of <40% for both As-S and As-Se. For air fills of 55% to 60%, a peak in gap area is seen, reflecting the number of wide closely spaced band gaps for air fills between 45% and 70% as demonstrated in Figs. 1(b) and 2(b) above. For air fills of 75%–80%, a minimum is seen, corresponding to the narrow band gaps crossing the air line as demonstrated in Figs. 1(c) and 2(c). Finally, for fills greater than 85%, the gap area increases dramatically as demonstrated by the wide band gaps crossing the air line that appear in Figs. 1(d) and 2(d) for the 90% air fill As-S and As-Se PBG fiber.
Figure 3 also shows the gap area of the largest band gap crossing the airline for silica triangular hole structure PBG fibers. We note that for fills <70%, the gap areas for As-S and As-Se based PBG fiber are comparable in magnitude to those of silica based fiber. For an air fill of 90%, the gap areas for As-S and As-Se based PBG fiber structures are approximately 0.25 the value of the silica based PBG fiber structure.
For the silica structure, the largest band gaps crossing the air line are high frequency band gaps that occur between bands 6 and 7, 8 and 9, and 12 and 13 for fills less than 70%. At fills of 70% and greater the low frequency band gap between bands 4 and 5 dominates. As air fill increases, the high frequency band gaps close and the low frequency band gap grows. Thus the general trend for silica is an increase in gap area with air fill as seen in Fig. 3. In contrast, the As-S and As-Se structures have a minimum in gap area at 75% to 80% air fill. Here, the band gaps crossing the air line which are responsible for the peaks around 55% to 60% air fill are high frequency band gaps. However, as these band gaps close with increasing air fill, the low frequency band gap seen in silica between bands 4 and 5 does not open and cross the air line until ~ 90% fill. Instead, the band gaps crossing the air line in As-S and As-Se for high air fills are high frequency band gaps which do not become appreciable until ~85% air fill. Consequently, a minimum in gap area is seen in Fig. 3 for the As-S and As-Se structures.
The large number of band gaps overlapping the air line in both the As-S and As-Se triangular structures allows much flexibility in fiber design. For example, consider a fiber designed using the large band gap crossing the air line centered at kΛ=10.6 in a 90% air fill As-S structure. For propagation of 4 µm light with kΛ=10.6 we find the center to center hole spacing to be 6.75 µm with a hole diameter of 6.72 µm.
We can use the approximate expressions of given by Cregan, et al.  to estimate the number of air-guided modes supported by the fibers
where NPBG is the number of guided modes, βH and βL are the propagation constants at the upper and lower edges of the band gap at a fixed frequency and n1 is the core index. The second equation, Eq. (1b) applies if k2 < .
For the 90% air fill As-S structure above with the central defect region formed by the removal of seven canes in a tube stacking fabrication process (rc =√7Λ/2), we find NPBG ~2.9. Thus we expect the fiber to guide at least a single transverse modes and possibly up to 3 modes at λ=4 µm. The calculated bound fundamental mode profile at kΛ=10.6 is shown in Fig. 4.
We have modeled As-Se and As-S triangular air-hole pattern PBG fibers. Several regions with large out of plane two-dimensional photonic band gaps are found for air fills >40% in both As-S and As-Se fibers. In particular, the structures have several wide and closely spaced band gaps at relatively low air fills of 45% to 60% that will ease fabrication of these fibers. Structures with high air fills of >85% exhibit very large band gaps that have potential for broadband IR laser transmission.
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