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The bend-insensitive lasing characteristics of a newly designed ytterbium-doped photonic crystal fiber (YPCF) are evaluated numerically. The designed YPCF remains single-mode and possesses large-mode-area of 1400 µm2 at 1064 nm wavelength with the beam quality factor (M 2) of 1.15, suggesting a diffraction-limited and continuous-wave lasing operation. The doped-region size is optimized for maximum conversion efficiency and it is found through numerical simulations that the doped radius should be more than 21 µm. The “mode expansion”, which is the self-expansion of the fundamental mode within the doped region with wavelength increments on bending the fiber, is the basic physical mechanism to give the bend-insensitive lasing performances of YPCF. It leads to an unusual variation of overlap factor when the wavelength is increased. A 41 cm long piece of YPCF demonstrates more than 83% of slope efficiency with 75% of conversion efficiency when pumped with a 975 nm laser source delivering an input power of 1 W.
©2008 Optical Society of America
1. Introduction
Lasers based on optical fibers are emerging as a potential solution for the vast range of applications in optical communications, industries, medical, and spectroscopy and promising to be alternatives to solid-state and semiconductor lasers due to their high reliability, thermal management, scalable output power, and high beam quality with narrow bandwidth. The Ytterbium (Yb)-doped fiber lasers operating from 1–1.117 µm wavelengths are the most fundamental laser source in optical systems as well for many industrial applications as Yb-doped silica fibers exhibit broad absorption and emission bands from ~800 nm to ~1064 nm for absorption and ~970 nm to ~1200 nm for emission [1]. Therefore, using a wide variety of pump sources (within absorption range), efficient lasing operation with narrow line width at any wavelength in the emission range can be achieved by using a set of fiber Bragg gratings that act as reflectors. One of the main concerns in designing the rare-earth doped fiber lasers is to alleviate the intensity dependent nonlinear effects such as Raman scattering and Brillouin scattering etc. In order to diminish the intensity-dependent nonlinear effects, a large fiber core should be used to attain a large mode area, while the use of a short fiber length increases the nonlinear threshold. In general, designing a large core generates the higher-order-modes (HOMs), so it becomes more important to realise a single mode operation with a large mode area and low bending losses. However, achieving a large mode area in conventional optical fibers is limited by the design space associated with them.
Photonic crystal fibers (PCFs) [2, 3], on the other hand, can be considered as alternatives to conventional optical fiber technology in many applications due to their robust single-mode propagation over a wide range of wavelengths [4], scalable modal areas and large degrees of freedom in designing their optical properties [5]. PCFs are composed of tiny air-holes in the cladding that run parallel to propagation direction with a hole-spacing Λ and hole-diameter d. The unusual optical characteristics of PCFs are governed by strongly wavelength dependent cladding index. Active dopants such as Erbium [6, 7] and Yb [8–13] have been introduced in the core of PCFs, realizing amplifiers and lasers. Furusawa et al. [9] demonstrated a 4.5 m long Yb-doped double clad PCF design lasing at 1.06 µm with 82% slope efficiency, whereas Dong et al. [12, 13] have shown the polarization-dependent and bend-resistant designs of large mode area, single-mode Ytterbium-doped PCFs (YPCFs) where the single-mode operation was achieved by enhancing leakage losses of the HOMs. Recently, Tsuchida et al. [14] have proposed a novel design of a passive PCF that exhibits low bending losses and a large mode area of 1400 µm2 at 1064 nm with improved beam quality factor (M 2=1.15) where the single-mode operation was achieved by coupling out the HOMs of the central core to the ring-core mode via resonant coupling mechanism.
In this paper, we have investigated the active properties of specially designed large-mode-area and single mode PCF [14] by introducing the Ytterbium ions into the core region. The most appealing feature of the YPCF design is the observance of the “expansion” of fundamental mode within the doped region upon bending the YPCF in a smaller bending radius when the wavelength is increased, instead of radiating out to outer cladding as experienced in conventional large mode area fibers. This “mode expansion” mechanism leads to the bend-insensitive lasing characteristics of YPCF. The slope efficiency of the YPCF almost remains constant when the fiber is subjected to bend in a smaller bending radius. The main cause that gives such interesting lasing properties was the unusual behaviour of the overlap factor. Our simulated results demonstrate an efficient YPCF laser operation at 1064 nm with more than 83% of slope efficiency in a short length (41 cm) of the fiber when the radius of the doping region is 25 µm. We have further predicted the size of doping region that can provides an excellent modal overlap, thus efficient laser operation with maximum conversion efficiency and high slope efficiency, even the fiber is bent into a 5 cm bending radius.
Fig. 1. A 3-D schematic of Ytterbium-doped PCF. The parameters r d, d, d 1, d 2, and Λ stand for the radius of the doped region, hole-diameters of small, large, medium air-holes, and the pitch constant of the PCF, are d/Λ=0.45, d 1/Λ=0.95, d 2/Λ=0.51, and Λ=20 µm.
2. Fiber design and its modal properties
Figure 1 illustrates the schematic of the fiber structure with doped region (shown by green circle) of radius r d. The other fiber parameters d, d 1, d 2, and Λ are d/Λ=0.45, d 1/Λ=0.95, d 2/Λ=0.51, and Λ=20 µm, where d, d 1, d 2, and Λ are the hole diameters and pitch-constant, respectively. The design strategy of the fiber structure is explained in our previous work [14]. The key feature of the design was to suppress the HOMs from the central core by matching the index of HOMs to the index of ring core mode at a particular wavelength, making HOMs leakier into the outer cladding whereas keeping the strong confinement of the fundamental mode that gives low bending losses and single mode operation at 1064 nm. The modal properties of the designed fiber are calculated accurately by implementing a full-vectorial finite element method (V-FEM) [15]. However, to compute the bending effect on the modal properties in a curved PCF, we have employed a V-FEM in local cylindrical coordinates system as described in Ref. [16].
Figure 2 shows the variation of effective mode area in a passive PCF [14] as a function of the bending radius for two wavelengths 975 nm (solid blue curve) and 1064 nm (solid red curve). It is evident from the graph that the effective mode area of both wavelengths closely follows each other and exhibits large effective mode area of 1400 µm2 when the PCF is bent into a larger bend loop, namely straight case. Thus, the threshold for nonlinear effects such as stimulated Raman scattering and Brillouin scattering can be increased and hence one can expects smaller fiber length for laser application. Note that, as the bending radius decreases, the effective mode area reduces that may result in a weak overlap between the doped region and the optical field.
Fig. 2. The variation of effective mode area of pump (975 nm, solid blue curve) and the output lasing wavelengths (1064 nm, solid red curve) as a function of bending radius in a passive PCF. It can be seen from the results that both curves closely follows each other as the bending radius is increased and attains nearly similar effective mode area of 1400 µm2.
3. Lasing characteristics of YPCF: the bending impact
In this section, we evaluate the transverse overlap between modal field and Ytterbium ions. It can be expressed in terms of overlap factor Γ and is calculated for pump and signal as [7]:
where Sd is the cross-section of the doped region, S is the cross-section of YPCF, and |Es,p|2 is the electric field intensity of the signal (s) and pump (p) waves, respectively. The overlap factor is computed for three different scenarios; (i) 10 µm doping radius (i.e. Λ/2), (ii) 15 µm doping radius, and (iii) 25 µm doping radius. We have evaluated the overlap factor in each case when the YPCF is straight, bent into 10 cm and 5 cm bending radii. In Figs. 3(a)–(c), we plot the spectral variation of overlap factor for straight fiber (solid green curve), 10 cm bending radius (dashed red curve), and 5 cm bending radius (dotted blue curve) for 10 µm (Fig. 3(a)), 15 µm (Fig. 3(b)), and 25 µm (Fig. 3(c)) doping radii, respectively. From these results, it can be examined that the value of overlap factor increases as the doping size is increased for straight fiber, and decreases as the wavelength increases (solid green curves). A peculiar behavior in overlap factor curves (dashed red and dotted blue lines) is noticed when the YPCF is bent into 10 cm and 5 cm bending radii. If we look to Fig. 3(a) for doping radius of 10 µm, the overlap factor increases as the wavelength increases (dashed red and dotted blue curve) which is exactly opposite to the trend observed in straight fiber case (solid green curve). However, the overlap factor decreases at a particular wavelength in the bent case in contrast to straight fiber. This is due to the shift of modal field on bending. The modal field tries to radiate out in the outer cladding region that decreases the strength of the overlap in the doped region thus reducing the overlap. It can be clearly seen that overlap of 97% can be obtained in a straight fiber case for 25 µm doped radius. If we further decrease the doping radius, the overlap decreases as most of the electric field doesn’t confine into the doped region as explained in Fig. 4 through modal field distribution.
Fig. 3. The overlap factor Γ as a function of wavelength for doping radius of (a) 10 µm, (b) 15 µm, and (c) 25 µm when the YPCF is straight (solid green curve), bended in 10 cm (dashed red curve) and 5 cm (dotted blue curve) bending radii. The overlap factor curves for bent cases show contrary behavior with increasing wavelength i.e. the overlap values increase as the wavelength increases on bending the YPCF due to the “mode expansion” mechanism that occurs when the YPCF is bent into smaller bending radius.
Fig. 4. The snapshots of the electric field distribution of fundamental mode at 1064 nm in YPCF when the fiber is straight (a), (d), (g); bent into 10 cm bending radius (b), (e), (h); 5 cm bending radius (c), (f), (i) for different doping radii r d=10 µm, 15 µm, and 25 µm, respectively.
In Fig. 4, we have displayed the electric field distribution of the fundamental mode at 1064 nm in YPCF for different doping radii with and without bending the fiber. If we look into Fig. 4(a), the modal field confines into the doped core (r d=10 µm). As we bend the YPCF in a 10 cm bending radius (Fig. 4(b)), most of the electric field remains in between the doped core and air-hole ring (i.e. in silica region) because on bending the fiber the field starts to spread out from the doped core. But, due to the presence of strong reflectors (i.e. air-holes), it confines in the silica region, thus reducing the modal overlap. The modal field strength reduces further from the doped core on decreasing the bending radius to 5 cm (as shown in Fig. 4(c) and dotted blue curve in Fig. 3(a)) and hence decreases the overlap value. If we increase the doping radius to 25 µm, as shown in Figs. 4(g)–4(i), more than 95% of the overlap value (see Fig. 3(c)) can be obtained when the YPCF is straight and above 75% as it is bent into 5 cm bending radius.
Fig. 5. The electric field distributions of fundamental mode in a 25 µm doping radius YPCF bent into a 10 cm bending radius at (a) 900 nm and (b) 1200 nm wavelengths. It can be clearly seen that the modal field expands within the doped region towards the center of the core as indicated by solid black arrow when the wavelength increases. This “mode expansion” mechanism gives unusual variation of the modal overlap which leads to the bend-insensitive lasing performance of the YPCF.
Fig. 6. Overlap factor as a function of the inverse of the bending radius at pump 975 nm (dashed blue curve) and the output lasing signal 1064 nm (solid red curve).
We refer this particular behavior as the “mode-expansion”, where the modal field starts self-expanding inside the doped region toward the center of the core as the wavelength increases that enhances the strength of the modal overlap, providing a growing overlap variation with increasing wavelength, which leads to the bend-insensitive lasing characteristics of YPCF. We can clearly visualize the “mode-expansion” mechanism in Figs. 5(a) and 5(b). The electric field distorts on bending the YPCF, but starts to expand as wavelength increases within the core and gives large overlap values as the doped region is enlarged to 25 µm radius. Figure 5(a) depicts the electric field distribution of fundamental mode in a 25 µm doping radius YPCF bended in a 10 cm bending radius at 900 nm wavelength. The modal field is deformed on bending the fiber and starts expanding towards the center of the core as the wavelength is increased to 1200 nm, as displayed in Fig. 5(b). The embedded arrow in Fig. 5(b) indicates the direction of the expansion of the modal field.
Figure 6 shows the variation of the overlap factor as a function of bend radius R bend inverse (1/R bend) in YPCF with 25 µm doping radius. The solid red curve dictates the overlap factor for 1064 nm wavelength which is our targeted output lasing wavelength, whereas the dashed blue curve stands for 975 nm wavelength (i.e. pump wavelength). From this graph, it is evident that the overlap for both pump and lasing signal is 97% when the fiber is straight and gradually decreases as the fiber is bent into smaller bending radius.
Table 1. The optical parameters for YPCF laser.
Table 1 lists the optical parameters used in determining the lasing characteristics of YPCF. The Yb ion meta-stable level life time τ is 0.8 ms, the absorption cross-section σa p=2.35×10-24 m2 at the pump wavelength λp, and σa s=2.95×10-27 m2 at the signal wavelength λs, the emission cross-sections are σe p=2.17×10-24 m2 and σe s=2.5×10-25 m2 at pump and signal, respectively. The Yb ions density in the core region N is 8×1024 ions/m3 (~1000 ppm). The power propagation equations [17] for forward pump, forward and backward signals are numerically solved by Runge-Kutta’s based algorithm in MATLAB [18] with appropriate boundary conditions required for lasing. Note that, we have neglected the amplified spontaneous emission, background losses, and excited state absorption in our numerical simulations. Additionally, any change in the index of silica by doping of Yb3+-ions is also neglected as the doping level of Yb3+ ions is not high. The overlap factor is then the overlap between the modal field and the doped region. If a high concentration of Yb3+ ions is used, then one has to consider the index increment caused due to Yb3+ ions as well other co-dopants such as Aluminum to increase the solubility of the ions into silica glass matrix.
In simulating the lasing characteristics, we have assumed fiber Bragg gratings (FBGs) laser cavity for a narrow line width and single-frequency operation. However, one can build up Fabry-Perot cavity, formed by using normal flat mirrors that may allow the emission of all lasing wavelengths. The usage of normal mirrors degrades the laser performances due to occurrence of the cavity losses. Therefore, to realize a single frequency operation, we have considered FBGs laser cavity. This would also ease the integration of all-fiber components and may reduce the losses incurred by non-fiber components.
3.1 Doped radius rd=10 µm
Figure 7(a) determines the length of YPCF with 10 µm doped radius, required in attaining the maximum conversion efficiency η. It shows the variation of fiber length as a function of reflectivity R out of the FBG for output lasing wavelength. An input pump power of 1W is used in the simulation. A length of 1.22 m is computed for maximum conversion efficiency of 75% when the R out is 90%. After deducing the YPCF length of 1.22 m, the lasing characteristics of the YPCF are obtained for different input pump powers. The lasing characteristics of 10 µm doped YPCF are presented in Fig. 7(b) for no bending and bending cases. The solid green curve corresponds to the output lasing power when the PCF is straight. The output power rises after the threshold and increases linearly as the input pump power increases. This gives the slope efficiency of 83%. As we bend the YPCF in 10 cm bending radius, the lasing power decreases due to small modal overlap value and thus provide low slope efficiency of 48%. However, the YPCF didn’t lase at all when it is bent into smaller bending radius of 5 cm. This is due to very weak confinement of the signal mode into the doped region.
Fig. 7. (a) The plot between fiber length L and the reflectivity R out of FBG corresponding to the lasing wavelength with conversion efficiency as a parameter when the input pump power is 1 W. The fiber length can be deduced from the above graph for maximum conversion efficiency. The length of YPCF is 1.22 m for maximum conversion efficiency of 75% at 90% of reflectivity of output FBG, (b) the output lasing characteristics of 1.22 m long YPCF as a function of input pump power when the YPCF is straight (solid green curve), bent into 10 cm (dashed red curve) and 5 cm (dotted blue curve) bending radii. The corresponding slope efficiencies are 83 %, 48%, and 0%, respectively.
3.2 Doped radius rd=15 µm
As we increase the doping radius to 15 µm, the length of the YPCF is get shorter and the confinement of the lasing signal as well as pump improves on bending the YPCF. Figure 8(a) demonstrates the color map between the fiber length and R out at an input pump power of 1 W. A length of 0.65 m is deduced from Fig. 8(a) for YPCF that has a 15 µm doped radius and the reflectivity of output FBG 90% for a maximum conversion efficiency of 75%. Using these parameters, the lasing characteristics of 15 µm doped YPCF are evaluated numerically and are shown in Fig. 8(b). The solid green curve stands for no bending case, whereas the dashed red and dotted blue curves represent 10 cm and 5 cm bending, respectively. The output lasing power increases linearly as a function of input pump power as expected. The YPCF furnish lasing power on bending 0.65 m long YPCF into 5 cm bending radius and exhibits a slope efficiency of 19%.
Fig. 8. (a) The plot between fiber length L and the reflectivity R out of FBG corresponding to the lasing wavelength with conversion efficiency as a parameter when the input pump power is 1 W. The fiber length can be deduced from the above graph for maximum conversion efficiency. The length of YPCF is 0.65 m for maximum conversion efficiency of 75% at 90% of reflectivity of output FBG, (b) the output lasing characteristics of 0.65 m long YPCF as a function of input pump power when the YPCF is straight (solid green curve), bent into 10 cm (dashed red curve) and 5 cm (dotted blue curve) bending radii. The corresponding slope efficiencies are 84 %, 81%, and 19%, respectively.
The output lasing power for 10 cm bending radius (dashed red curve) closely follows the solid green curve that correspond for no bend case, which shows a slope efficiency of 81% that is near to the slope efficiency of 84% of straight fiber case. One can realize from these results that on bending the YPCF in 10 cm bending radius, the lasing characteristics almost remain similar, which is one of the merit of using specially designed YPCF.
3.3 Doped radius rd=25 µm
It has been pointed out in previous sub-section 3.2 that by enlarging the doping size, the length of YPCF can be shortened as well one can realize similar lasing characteristics for straight and bent cases. Therefore, we further increase the doping radius to 25 µm which can be estimated as the upper limit for extending the doping region, feasible to fabricate the YPCF. The fiber length is computed from the contour graph shown in Fig. 9(a) where the fiber length is plotted versus the reflectivity of output FBG as a function of conversion efficiency η, when the input pump power is 1 W. The YPCF length is calculated as 0.41 m, which is small in comparison to all published Yb-doped PCF lasers [8–13], for a conversion efficiency of 75% and 90% of output FBG reflectivity. After determining the YPCF length, we obtain its lasing characteristics as shown in Fig. 9(b). The solid green curve illustrates the straight case, while the dashed red and dotted blue curves signify 10 cm and 5 cm bending. The output lasing characteristics in straight fiber case increases linearly as expected and deliver a slope efficiency of 83%, while the slope efficiency increase a little when the YPCF is bent into 10 cm bending radius, furnishing 85% of slope efficiency and 86% of slope efficiency when the YPCF is bent into 5 cm bending radius. Two important conclusions can be made from these results. First, the lasing characteristic curves closely follows each other for straight and bending cases, thus providing almost constant slope efficiency of the YPCF laser, allowing the bend-insensitive lasing operation. And secondly, the lasing curves are interchanged contrary to 15 µm doped YPCF example. We may finally conclude that by enlarging the doped region in YPCF with low bending loss performance, an efficient lasing operation with bend-resistant characteristics can be realized, attributed to “mode expansion” mechanism.
Table 2 sums up the lasing characteristics of YPCF for different doping radii. It is apparent that a 41 cm long piece of YPCF possessing 25 µm doping radius delivers almost similar lasing characteristics even the YPCF is bent into a smaller bending radius of 5 cm.
Fig. 9. (a) The plot between fiber length L and the reflectivity R out of FBG corresponding to the lasing wavelength with conversion efficiency as a parameter when the input pump power is 1 W. The fiber length can be deduced from the above graph for maximum conversion efficiency. The length of YPCF is 0.41 m for maximum conversion efficiency of 75% at 90% of reflectivity of output FBG, (b) the output lasing characteristics of 0.41 m long YPCF as a function of input pump power when the YPCF is straight (solid green curve), bent into 10 cm (dashed red curve) and 5 cm (dotted blue curve) bending radii. The corresponding slope efficiencies are 83 %, 85%, and 86%, respectively.
Table 2. Summary of the bend-insensitive lasing characteristics of YPCF
Fig. 10. Comparison between two approaches used to obtain lasing performances of the proposed YPCF. The solid red curve corresponds to the laser output power calculated from the analytical relations mentioned in Ref. [19], whereas the solid blue curve stands for the laser characteristics obtained numerically by solving the rate equations. A 7% of error is estimated in slope efficiencies between two curves.
Fig. 11. Slope efficiency as a function of doped radius r d. On vertical axis, the difference of slope efficiencies (ΔS=S 5cm-S straight, where S 5cm and S straight are the slope efficiencies when the YPCF is bent in 5 cm bending radius and it is kept straight) is plotted. It is apparent from the numerical result that a high and an almost constant slope efficiency can be obtained if the doped radius is assumed larger than 21 µm i.e. r d≥21 µm.
In Fig. 10, we show the comparison between lasing performances of a 25 µm doped YPCF calculated using analytical [19] (solid red curve) and numerical (solid blue curve) approaches. Analytical relations have been used from Ref. [19], where the linear laser cavity is built by reflecting and transmitting normal mirrors. To obtain the lasing characteristics by analytical formulas, the intra-cavity losses have been neglected. The lasing output power varies linearly with input pump power and exhibits a slope efficiency of ~90%, whereas, the slope efficiency computed through numerical simulations of the rate equations is ~83 %, resulting an error of approximately 7%, which we believe due to the exact solutions of the rate equations as mentioned in Ref. [19]. In both analytical and numerical approaches, the saturation effects of the active ions are considered by defining the saturation pump power [1], cross-saturation, and intrinsic saturation coefficients [19]. The saturation pump power is obtained as 110.7 mW and the critical pump power [1] is ~3 mW.
Fig. 12. The contour plot between the overlap factors for pump Γp and lasing signal Γs with output lasing power P o as a parameter for 41 cm long YPCF with 25 µm doped radius. It can be observed from the graph that Γs>Γp when the YPCF is bent, which is opposite to what is found in conventional Yb-doped fibers and PCFs. The larger value of Γs on bending can lead to high powers from YPCF laser. Also, note that the output power P o increases on shortening the bending radius.
4. Discussion and concluding remarks
The lasing characteristics of YPCF for different doping radii are illustrated numerically and well described in Section 3. We further explain the possibility of achieving high powers by employing the PCF structures and actively doping its core as well also formulate for deriving the larger doping size. Figure 11 describes the slope efficiency variation as a function of the doped radius. The difference of slope efficiencies ΔS, which is defined as the difference between the slope efficiency of 5 cm bending case (S 5cm) and the slope efficiency of no bending case i.e. straight fiber (S straight), can be expressed as ΔS=S 5cm-S straight. It can be revealed from Fig. 10 that ΔS is very less for smaller doped radius and improves as the doped radius is enlarged (as shown by solid blue curve). Consequently, ΔS is stabilized after a doping radius of 21 µm, approaching to almost zero value. A very interesting and important conclusion can be made from this result that the slope efficiency becomes constant for both straight and bending cases, resulting into the bend-insensitive lasing operation of YPCF.
Figure 12 depicts the contour plot between overlap factors for pump (Γp) and lasing signal (Γs) in a 41 cm long YPCF, which has 25 µm doping radius, as a function of output power, when the input pump power is 1W. The dark red region in the contour map shows the maximum output power, while the dark blue region corresponds to less output power. The output lasing power P o has also been plotted by open circles for no bending and bending of YPCF at 10 cm and 5 cm bending radii. It can be anticipated from this graph that the overlap factor for signal becomes larger than the overlap factor for pump (i.e. Γs>Γp) when the YPCF is bent into 10 cm and 5 cm bending radii. Another important conclusion that can be made from this result is that the output power increases slowly by bending the YPCF. This can provide a possibility of realizing high powered lasers using YPCFs.
To summarize our work, we have numerically obtained the lasing characteristics of specially designed YPCF, whose specific design leads to bend-insensitive lasing performances. The “mode expansion” mechanism within the doped core region was noticed. And it was discovered that the overlap factor increases as the wavelength increases on bending the YPCF, contrary to the effect where overlap factor decreases as the wavelength increases in general. This peculiar characteristic of overlap allows the high power of lasing signal even on bending the fiber.
Further, numerical simulations reveal that almost constant slope efficiencies can be delivered for 1064 nm lasing wavelength when the YPCF is kept straight and is bent into smaller bending radii of 10 cm and 5 cm with enlarged doping region maintained larger than 21 µm. A slope efficiency of 83% was observed in a 41 cm long YPCF when it was straight, while slope efficiencies of 85% and 86% were obtained when the YPCF was bended into 10 cm and 5 cm bending radii, respectively. We expect that our numerical study on designing the bend-insensitive lasing operation of YPCF would be applicable to industrial applications. We also look forward for the practical realization of our proposed single-mode and large mode area YPCF lasers.
Acknowledgements
S.K. Varshney kindly acknowledges to Japan Society for Promotion of Science (JSPS) for their support in carrying out this work. Authors also acknowledge to Indo-Japan collaboration project on: Infrastructural Technology for Advanced Use of Photonic Crystal Fibers in Optical Communication Systems.
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