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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 26 — Dec. 24, 2007
  • pp: 17724–17735
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Dispersion, birefringence, and amplification characteristics of newly designed dispersion compensating hole-assisted fibers

Kunimasa Saitoh, Shailendra K. Varshney, and Masanori Koshiba  »View Author Affiliations


Optics Express, Vol. 15, Issue 26, pp. 17724-17735 (2007)
http://dx.doi.org/10.1364/OE.15.017724


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Abstract

We propose a new design of hole-assisted fiber (HAF) that can compensate for the accumulated dispersion in single-mode fiber link along with dispersion slope, thus providing broadband dispersion compensation over C-band as well as can amplify the signal channels by utilizing the stimulated Raman scattering phenomena. The proposed dispersion-compensating HAF (DCHAF) exhibits the lowest dispersion coefficient of -550 ps/nm/km at 1550 nm with an effective mode area of 15.6 µm2. A 2.52 km long module of DCHAF amplifies incoming signals by rendering a gain of 4.2 dB with ±0.8 dB gain flatness over whole C-band. To obtain accurate modal properties of DCHAF, a full-vector finite element method (FEM) solver is employed. The macro-bend loss characteristics of the proposed DCHAF are evaluated using FEM solver in cylindrical coordinate systems of a curved DCHAF, and low bending losses (<10-2 dB/m for 1 cm bending radius) are obtained for improved DCHAF design while keeping intact its dispersion compensation and Raman amplification properties. We have further investigated the birefringence characteristics that can give significant information on the polarization mode dispersion of DCHAF by assuming a certain deformation (eccentricity e=7%) either in air-holes or in the doped core or in both at a same time. It is noticed that the distortion in air-holes induces a birefringence of 10-5, which is larger by a factor of 10 than the birefringence caused due to the core ellipticity. A PMD of 11.3 ps/√km is obtained at 1550 nm for distorted air-holes DCHAF structure.

© 2007 Optical Society of America

1. Introduction

Recent developments in optical fiber science have led to the birth of photonic crystal fibers (PCFs) [4

4. P.St.J. Russell, “Photonic crystal fiber,” Science 288, 358–362 (2003). [CrossRef]

6

6. A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibres (Kluwer Academic, The Netherlands, 2003). [CrossRef]

], where the micro-sized air-holes running down in the propagation direction build the cladding. These microstructured fibers are gaining significant interest due to their novel optical properties such as wide-band single-mode behavior [7

7. T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett.22, 961–963 (1997).

], large mode area, high nonlinearity, small effective mode area, and manageable dispersion characteristics [6

6. A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibres (Kluwer Academic, The Netherlands, 2003). [CrossRef]

, 8

8. K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fibers: Application to ultra-flattened dispersion,” Opt. Express 11, 843–852 (2003). [CrossRef] [PubMed]

]. Several designs of PCFs based on dual-concentric core refractive index profile have been studied to discover dispersion compensating characteristics along with Raman amplification [9

9. S.K. Varshney, K. Saitoh, and M. Koshiba, “A novel design for dispersion-compensating photonic crystal fiber Raman amplifier,” IEEE Photon. Technol. Lett. 17, 2062–2064 (2005). [CrossRef]

15

15. B.J. Mangan, F. Couny, L. Farr, A. Langford, P.J. Roberts, D.P. Williams, M. Banham, M.W. Mason, D.F. Murphy, E.A.M. Brown, H. Sabert, T.A. Birks, J.C. Knight, and P.St.J. Russell, “Slope-matched dispersion-compensating photonic crystal fibre,” in Proceedings of Conference on Lasers and Electro-Optics (CLEO 2004), paper CPDD3, San Francisco, CA, (2004). [PubMed]

]. The main drawback in dispersion compensating PCF designs was the adoption of large number of air-hole rings in the cladding that can give rise to enormous Rayleigh scattering losses, thus large optical attenuation. In order to have a large negative dispersion coefficient with low scattering losses, we examine a variant of PCF, namely, hole-assisted fiber (HAF) which has a germanium-doped core surrounded by either one or two layers of air-hole rings [16

16. T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss,” Opt. Express 9, 681–686 (2001). [CrossRef] [PubMed]

, 17

17. K. Saitoh, Y. Tsuchida, and M. Koshiba, “Bending-insensitive single-mode hole-assisted fibers with reduced splice loss,” Opt. Lett. 30, 1779–1781 (2005). [CrossRef] [PubMed]

].

In this work, we propose a simpler and intriguing design of dispersion compensating HAF (DCHAF), where only one air-hole ring composed of six air-holes surrounds a germanium-doped core. The proposed DCHAF shows a large negative dispersion coefficient of -550 ps/nm/km at 1550 nm, almost two to five times larger in magnitude than conventional broadband DCF with compatible and relatively large effective mode area of 15.6 µm2 at 1550 nm wavelength. To emphasize the potential of the proposed DCHAF as a discrete Raman amplifier, we have also modeled its Raman properties and found that DCHAF yields a high Raman gain efficiency of 3.4 W-1·km-1 at 12.6 THz frequency shift and provides a gain of 4.2 dB with gain ripple of ±0.8 dB in the entire C-band when a 2.52 km module of DCHAF is pumped by a 1450 nm single pump source with 560 mW pump power in the backward direction. Further, we have improved the DCHAF design by introducing an additional outer air-hole ring with large air-hole diameter to reduce the bending losses below 0.01 dB/m at short bending radius without affecting its dispersion and amplification properties. Finally, we have numerically computed the polarization mode dispersion (PMD) of a DCHAF by assuming certain deformation (characterized through eccentricity of an ellipse) in the air-holes and the doped core. The presence of deformation in the fiber geometry (air-holes and doped core) induces a birefringence, which is of the order of 10-5 in the case of air-holes distortion while 10-6 in the doped elliptical core oriented vertically. The PMD is obtained by Jones matrix eigen analysis [18

18. B.L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992). [CrossRef]

] by assuming random mode coupling between polarization modes. A PMD of 11.3 ps/√km is achieved at 1550 nm for distorted air-holes DCHAF.

2. Fiber design and dispersion properties

Fig. 1. (a) Schematic of the germanium-doped HAF that has surrounding six air-holes with hole diameter d=0.35 µm, spaced at a pitch constant Λ=1.3 µm. The doped radius a is 1.085 µm, (b) equivalent refractive index profile of the proposed DCHAF, where Δ+ and Δ- show the raising and lowering index deltas, respectively. The raising index delta Δ+, achieved via germanium doping, is set to 3.0%, whereas the Δ- is determined by air-holes encircling the doped core. The background material is pure silica glass.
Fig. 2. (a) The optical field intensity distribution in DCHAF at 1550 nm wavelength. (b) Effective index variation of cladding (solid red curve), fundamental mode (solid blue curve), and doped-core (solid green curve), (c) the magnified view of the effective indices variation of cladding and fundamental mode. It can be clearly seen from the numerical results that the index of the fundamental mode approaches to the cladding index as the wavelength increases and becomes smaller after a particular wavelength, referred as phase matching wavelength, which is approximately 1.7 µm in DCHAF.

The variation of effective indices of the fundamental mode (solid blue curve), the doped-core (solid green curve), and the cladding (solid red curve) is depicted in Fig. 2(b) and the magnified view of the fundamental mode and cladding indices variation is shown in Fig. 2(c). It is evident from Fig. 2(b) that the effective index of the fundamental mode approaches to the cladding index as wavelength increases and becomes smaller after a particular wavelength, which is referred as the phase matching wavelength. In the case of DCHAF, the phase matching wavelength is obtained around 1.7 µm. The DCHAF shows a cut-off wavelength at 840 nm for the second-order mode (LP11-like mode), which makes it single mode for desired range of wavelengths (i.e. over communication range wavelengths). We define the cutoff wavelength where the effective index of the second-order mode becomes equal to the pure silica index.

Fig. 3. (a) Chromatic dispersion of the optimized DCHAF (solid blue curve) and the residual dispersion (solid green curve) left behind after the dispersion compensation of 80 km SMF link by 2.52 km DCHAF, (b) the impact of ±2% tolerances present either in the pitch constant or the hole-diameter. The DCHAF exhibits a largest negative dispersion coefficient of -552 ps/nm/km at 1550 nm. It is found from tolerance analysis that chromatic dispersion may change by ±12% and ±18% for ±2% tolerances in pitch constant and hole-diameter, respectively. However, the RDS of the DCHAF with tolerances can be matched with the RDS of conventional SMF.
Fig. 4. (a) Macro-bending loss characteristics of DCHAF at a wavelength of 1550 nm as a function of bending radius with and without additional air-hole ring, as depicted on right, (b) schematic of the improved DCHAF structure, where an additional air-hole ring is introduced in the outer cladding region with large air-hole diameter d’=12 µm and air-hole spacing Λ’=15 µm. It is apparent from the results that the DCHAF without an extra air-hole ring suffer from large bending losses at smaller bending radii, however, the bending losses reduce sharply when an additional air-hole ring is introduced, which lowers the bending losses below 0.01 dB/m at 1 cm bending radius. Surprisingly, it is also detected that the dispersion properties of DCHAF don’t change by the presence of an extra air-hole ring.

Further, we have investigated the impact of fiber tolerances on the dispersion property of DCHAF, as displayed in Fig. 3(b). The solid blue curve stands for the optimized chromatic dispersion (as shown in Fig. 3(a)), while the dashed and solid red curves correspond to ±2% deviation in the pitch constant (ΔΛ), and the solid and dashed green curves indicate ±2% variation in the hole-diameter (Δ d), respectively. The presence of fiber tolerances up to ±2% may affect its dispersion characteristics. The absolute value of the chromatic dispersion is changed by ±12% and ±18% for a ±2% variation in the pitch-constant Λ and the air-hole diameter d, respectively. It is found that even if the structural parameters are changed up to ±2%, the RDS of the DCHAF can be matched with that of conventional SMF.

Next, we have calculated macro-bending loss characteristics of the DCHAF. To compute the bending losses even at shorter bending radius of 1 cm, we have carried out an accurate analysis of a curved DCHAF by using a newly developed approach based on cylindrical coordinate systems in full-vectorial FEM solver, as described in Ref. [20

20. K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express 14, 11128–11141 (2006). [CrossRef] [PubMed]

]. The macro-bending loss characteristics of DCHAF are presented in Fig. 4(a) shown by solid blue curve, where the operating wavelength is 1550 nm. It can be revealed from the results that the bended DCHAF in a 5 cm bending radius offers bending loss lower than 1 dB/m, but the bending losses increases as the bending radius is decreased and the DCHAF shows bending losses more than 100 dB/m for 1 cm bending radius. In order to reduce the bending losses sharply, we have introduced an additional air-hole ring, as shown in Fig. 4(b), with larger air-holes diameter d′=12 µm with air-hole spacing Λ’=15 µm. Surprisingly, it was discovered that the bending losses drop drastically, lower than 0.01 dB/m for a 1 cm bending radius, as shown by solid red curve in Fig. 4(a). The bending loss at longer wavelength of 1565 nm is almost at same level as that of 1550 nm and is below 0.01 dB/m for bending radius larger than 1.1 cm.

We have also simulated the improved DCHAF structure (as depicted in Fig. 4(b)) and obtain its dispersion characteristics. It is observed that the chromatic dispersion is not affected by the presence of additional air-hole ring in outer cladding. Therefore, the key role of inserting an extra air-hole ring is just to reduce the bending losses. Even if we add a second ring of holes far away in the cladding, the effective index of the second-order mode remains equal to the pure silica index at 840 nm wavelength, while in this modified structure, the second-order mode exists as a leaky mode when its effective index becomes smaller than the silica index. However, we have confirmed that the leakage loss level of the second-order mode is higher than 150 dB/m at 840 nm, therefore, we can consider the improved DCHAF structure with additional air-hole ring as effectively single mode above 840 nm wavelength range.

3. Raman amplification characteristics of DCHAF module

In this section, we evaluate amplification properties of a lumped DCHAF by employing the stimulated Raman scattering phenomenon. A theoretical background to compute the Raman gain efficiency (RGE) γ R and Raman gain is well described in Refs. [9

9. S.K. Varshney, K. Saitoh, and M. Koshiba, “A novel design for dispersion-compensating photonic crystal fiber Raman amplifier,” IEEE Photon. Technol. Lett. 17, 2062–2064 (2005). [CrossRef]

, 10

10. S.K. Varshney, T. Fujisawa, K. Saitoh, and M. Koshiba, “Novel design of inherently gain-flattened discrete highly nonlinear photonic crystal fiber Raman amplifier and dispersion compensation using a single pump in C-band,” Opt. Express 13, 9516–9526 (2005). [CrossRef] [PubMed]

, and 21

21. M. Bottacini, F. Poli, A. Cucinotta, and S. Selleri, “Modeling of photonic crystal fiber Raman amplifiers,” J. Lightwave Technol. 22, 1707–1713 (2004). [CrossRef]

]. Therefore, we don’t present all basic mathematical formulation here. As an initial step to assess the Raman amplification performances of the DCHAF, we obtain the RGE and Rayleigh backscattering coefficient. The spectral variation of RGE is depicted in Fig. 5(a), as shown by dotted blue curve. Additionally, we have also plotted the effective mode area A eff variation in the same graph exhibited by solid red curve. A peak RGE of 3.4 W-1·km-1 is obtained at a frequency shift of 12.6 THz. The DCHAF shows a relatively large effective mode area of 15.6 µm2 at 1550 nm wavelength, almost equal to a standard DCF’s mode area.

After determining the primary parameters; RGE and Rayleigh backscattering coefficient, we solve the rate equations governing the power evolution of pump and signal powers along the length of the fiber. We have assumed a background loss of 1.4 dB/km at 1550 nm [22

22. M. Onishi, Y. Koyano, M. Shigematsu, H. Kanamori, and H. Nishimura, “Dispersion compensating fiber with a high figure of merit of 250 ps/nm/dB,” Electron. Lett. 30, 161–163 (1994). [CrossRef]

] with its spectral variation in numerical simulations. The signal power in the forward direction is obtained at the end of the fiber length and hence the Raman gain can be determined by the ratio value between the signal power at the end of fiber length and input signal power. The 2.52 km long DCHAF module is mono-pumped in the backward direction, whereas the signal channels, equally spaced by 200 GHz with an initial power of -10 dBm/ch are launched in the forward direction. The pump power and wavelength are selected as 560 mW and 1450 nm, respectively. The spectral variation of Raman gain is demonstrated in Fig. 5(b). A maximum gain of 4.2 dB is achieved with ±0.8 dB gain ripples over C-band. The DCHAF Raman amplifier exhibits noise figure of 9 dB, optical signal to noise ratio (OSNR) of 42.3 dB, and double Rayleigh backscattering of -57 dB on average. We may finally conclude that the proposed DCHAF not only compensates for the dispersion, but also amplifies the signals by providing a reasonable gain of 4.2 dB with low noise figure, high OSNR, and very low double Rayleigh backscattering.

Fig. 5. (a) Spectral variation of RGE γ R (dotted blue curve) and effective mode area A eff (solid red curve). A peak RGE of 3.4 W-1·km-1 is obtained at 12.6 THz frequency shift and a 15.6 µm2 of effective mode area is found at 1550 nm wavelength, (b) variation of Raman gain as a function of the wavelength when a 2.52 km long DCHAF module is singly pumped in the backward direction. A peak gain of 4.2 dB is incurred with ±0.8 dB gain ripples over C-band. The DCHAF Raman amplifier exhibits a 9 dB of noise figure, a 42 dB of OSNR, and a -57 dB double Rayleigh backscattering values on average.

4. Birefringence and PMD calculations

In this section, we examine the birefringence properties of DCHAF by assuming the presence of certain deformations either in air-holes or in the doped core that may be caused during the fabrication of the fiber. These unwanted distortions in the air-holes or in the core may induce the birefringence to fiber. The deformation is characterized in terms of eccentricity of an ellipse, which is given by:

eh=1b2a2a>b
eV=1a2b2b>a
(1)

where 2a and 2b are the length of major or minor axes of an ellipse, respectively, the subscripts h and V stand for ellipse orientation to x- and y-axes, respectively. We consider several cases such as core remain circular while the air-holes are elongated either horizontally or vertically or a vice-versa. Figures 6(a)(h) illustrate different scenarios of possible deformations in the air-holes or in the core that may occur in DCHAF while fabrication. We have considered all possible combinations, for example, circular core with distorted air-holes (Figs. 6(a) and (b)), deformed doped-core with circular air-holes (Figs. 6(c) and (d)), both doped-core and air-holes are deformed (Figs. 6(e)(h)). However, the distortion that may arise in air-holes or in the core is on the order of a few nm [23

23. Through e-mail correspondence with crystal-fiber company (www.crystal-fiber.com).

]. We define this distortion in terms of the eccentricity caused either to air-holes or in the core according to Eq. (1). An eccentricity of 7% (i.e. e=0.07) is assumed in all cases, similar to the value considered by Millot et al. in Ref. [24

24. G. Millot, A. Sauter, J.M. Dudley, L. Provino, and R.S. Windeler, “Polarization mode dispersion and vectorial modulational instability in air-silica microstructure fiber,” Opt. Lett. 27, 695–697 (2002). [CrossRef]

].

Fig. 6. Cross-sections of DCHAF when a certain distortion (e=7%) presents in air-holes or in the core; the air-holes are stretched along (a) x-axis, (b) y-axis, the core is elongated along (c) x-axis, (d) y-axis, (e) both core and air-holes are extended along x-axis (f) core is elongated along x-axis while the air-holes along to y-axis, (g) core is elongated along y-axis whereas air-holes along to x-axis, (h) both core and air-holes are stretched along y-axis.

The birefringence for all structures (Figs. 6(a)(h)) is computed through full-vectorial FEM. We plot the birefringence as a function of wavelength in Figs. 7(a) and (b). In Fig. 7(a), the solid curves correspond to the distortion in the air-holes only, where the core shape is perfect circular, while the dotted curves stand to the deformation in the core, where the surrounding air-holes are perfectly circular. The blue curves indicate the case when the air-holes or core are elongated horizontally, whereas the red curves suggest the vertical orientation of air-holes or core. It can be seen that the deformation in air-holes induces higher birefringence than the birefringence due to core ovality. The DCHAF exhibits a birefringence of the order of 10-6 when the core is oval, while birefringence increases by a factor of 10 when the air-holes are deformed. However, both the core and air-holes are distorted at the same time as shown by Figs. 6(e)6(h), the birefringence increases slightly as shown in Fig. 7(b), e.g. in case of 6(b) DCHAF, the birefringence at 1550 nm is 1.2×10-5, whereas in 6(h) DCHAF it is 1.4 ×10-5. This high birefringence in DCHAF may give rise to large PMD values.

The influence of increased eccentricity is also studied and the numerical results are shown in Fig. 8(a). The red and blue curves correspond to horizontal and vertical orientation of distorted air-holes, while the core is kept circular. The solid and dashed lines dictate 7% and 40% eccentricity, respectively. It can be clearly examined from the graph that the birefringence jumps by a factor of 10 for an eccentricity of 40% in air-holes only. However, it’s uncommon to have such a large value of eccentricity in practice. It is also noted that the parameter’s distortion in DCHAF may affect the dispersion properties. Therefore, we have investigated the impact of distortion on dispersion characteristics of the optimized DCHAF for deformations illustrated in Figs. 6(a)(d). The results are embedded in Fig. 8(b). The solid blue curve corresponds to the dispersion characteristics of optimized DCHAF structure as described in Fig. 1(a), while other solid and dashed curves stand for the x- and y-polarization of the fundamental mode of respective structures. The solid, dashed red and black curves indicate for x- and y-polarizations of the fundamental mode in 6(a) and 6(b) DCHAF structures, respectively, whereas the solid, dashed green and magenta curves correspond to DCHAF structures shown in Figs. 6(c) and 6(d), respectively. It is evident from the numerical results that the distortion in air-hole shifts the dispersion coefficient at 1550 nm by a ±2% around the optimized value, however, the deformation in core wobbles the dispersion value by a ±0.5%.

Fig. 7. Birefringence as a function of wavelength in DCHAF (a) for structures 6(a)–6(d) and (b) for structures 6(e)–6(h). It can be deduced that the deviation in air-holes imparts a larger birefringence than the presence of ovality in the core, almost higher by a factor of 10. It is also apparent that the presence of distortion in both the core and air-holes at the same time enhances the birefringence slightly.
Fig. 8. (a) Comparison of the induced birefringence for two different eccentricities 7% and 40%, present in air-holes only. The birefringence increases by a factor of 10 when the air-holes get more elliptical as shown by dotted blue and red curves. (b) Impact of parameter’s distortion on the dispersion properties of optimized DCHAF. The solid and dotted curves correspond to the x- and y-polarization of the fundamental mode. It can be deduced from the results that the distortion in air-holes shifts the dispersion coefficient at 1550 nm by ±2%, whereas the deformation in core changes the dispersion coefficient by ±0.5%.

Finally, we evaluate the PMD in DCHAF. PMD, a serious concern in high bit rate systems (>10 Gb/s) employing optical fibers, arises due to asymmetries in the fiber core that induce a small amount of birefringence which randomly varies along the length of the fiber. This birefringence causes the power in each optical pulse to split between the two polarization modes of the fiber and travel at different speeds, creating a differential group delay (DGD) between the two modes that result in pulse spreading, thus degrading the capacity of the system [18

18. B.L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992). [CrossRef]

, 25

25. A.O. Dal Forno, A. Paradisi, R. Passy, and J.P. von der Weid, “Experimental and theoretical modeling of polarization-mode dispersion in single-mode fibers,” IEEE Photon. Technol. Lett. 12, 296–298 (2000). [CrossRef]

27

27. D. Gupta, A. Kumar, and K. Thyagarajan, “Polarization mode dispersion in single mode optical fibers due to core-ellipticity,” Opt. Commun. 263, 36–41 (2006). [CrossRef]

]. In case of PCFs, where tiny air-holes build the cladding instead of uniform low refractive index material, may exhibit slightly larger birefringence than conventional fibers, as demonstrated before in DCHAF case. To evaluate the PMD, we have adopted Jones matrix approach where random mode coupling between two polarization modes is taken into account. The fiber of length L is assumed as a concatenation of N number of segments with fixed birefringence axes in every segment [25

25. A.O. Dal Forno, A. Paradisi, R. Passy, and J.P. von der Weid, “Experimental and theoretical modeling of polarization-mode dispersion in single-mode fibers,” IEEE Photon. Technol. Lett. 12, 296–298 (2000). [CrossRef]

27

27. D. Gupta, A. Kumar, and K. Thyagarajan, “Polarization mode dispersion in single mode optical fibers due to core-ellipticity,” Opt. Commun. 263, 36–41 (2006). [CrossRef]

]. At each coupling site the relative alignment of the axes can drift with respect to each other due to variations in environmental conditions. As a result, the instantaneous value of DGD exhibited by the concatenated series of fiber sections may change. The Jones matrix T i for the ith section can be given as [27

27. D. Gupta, A. Kumar, and K. Thyagarajan, “Polarization mode dispersion in single mode optical fibers due to core-ellipticity,” Opt. Commun. 263, 36–41 (2006). [CrossRef]

]:

Ti=(cosθisinθisinθicosθi)(exp(jπBhiλ)00exp(jπBhiλ))(cosθisinθisinθicosθi)
(2)

where θi is the orientation of the fast axis of the ith segment with respect to a fixed axis, h i is the length of the ith segment, λ is the wavelength, and B is the birefringence computed as |n x-n y|. The Jones matrix for entire length of an optical fiber can be expressed as:

T=i=1NTi
(3)

To evaluate the DGD, Jones matrix eigen-analysis is used [18

18. B.L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992). [CrossRef]

], where Jones matrix of the fiber is calculated for two nearby wavelengths and the DGD, Δτ, is therefore given by:

Δτ(λ1)=arg(ρ1ρ2)dω
(4)

where ρ1 and ρ2 are the eigen-values of the matrix product ∂ω T(ω)T -1(ω), arg() is the argument function and dω is the change in optical angular frequency over two near wavelengths. The average value of DGD, <Δτ>, is known as the PMD of the fiber. The probability distribution (PD) of DGD is obtained by calculating Δτ large number of times by considering θi as a random quantity uniformly distributed in [0, 2π] and h i to be a random quantity chosen from Gaussian distribution around the mean length given as a ratio of fiber length to number of segments with a 20% of standard deviation [25

25. A.O. Dal Forno, A. Paradisi, R. Passy, and J.P. von der Weid, “Experimental and theoretical modeling of polarization-mode dispersion in single-mode fibers,” IEEE Photon. Technol. Lett. 12, 296–298 (2000). [CrossRef]

, 27

27. D. Gupta, A. Kumar, and K. Thyagarajan, “Polarization mode dispersion in single mode optical fibers due to core-ellipticity,” Opt. Commun. 263, 36–41 (2006). [CrossRef]

]. The DGD probability distribution Pτ) can be given in terms of Maxwellian distribution as:

P(Δτ)=32π2Δτ2Δτ3exp(4Δτ2πΔτ2)
(5)

where <Δτ> is the mean DGD. The mean PMD can be calculated as [27

27. D. Gupta, A. Kumar, and K. Thyagarajan, “Polarization mode dispersion in single mode optical fibers due to core-ellipticity,” Opt. Commun. 263, 36–41 (2006). [CrossRef]

]:

DPMD=ΔτL
(6)

The probability distribution for DGD is computed for structures shown in Figs. 6(a)(d) and is plotted in Fig. 9(a). We have considered 1000 DGD points and the average DGD for a 2.52 km long 6(a)–6(d) DCHAF structures is obtained as 100.84 ps, 101.58 ps, 99.4 ps, and 98.8 ps, respectively at 1550 nm wavelength. It is evident from the PD that the DCHAF with vertically oriented elliptical air-holes shows larger DGD than the other structures, hence it results into high PMD values. In Fig. 9(b), we present the PD for 6(a) DCHAF structure as a function of wavelength. The wavelength increases in a step of 5 nm beginning from 1530 nm from solid blue curve to solid black curve at 1560 nm. The peak of the PD shifts to left as wavelength decreases. In Fig. 10, we plot the mean PMD calculated through Eq. (6) when a large length (80 km) of a DCHAF (with horizontally oriented elliptical air-holes and circular core i.e. 6(a)) is selected. It is also confirmed from numerical simulations that if the length of DCHAF is shorter e.g. few kms, the average value of DGD doesn’t change almost. It can be concluded from the results shown in Fig. 10 that the DCHAF exhibits larger PMD values for longer wavelengths. The DCHAF may show a PMD of 11.36 ps/√km at 1550 nm wavelength for distorted air-holes stretched horizontally, whereas for deformed core elongated vertically may show a PMD of 11.16 ps/√km at 1550 nm.

Fig. 9. Variation of Maxwellian probability distribution Pτ) as a function of DGD (a) in various DCHAF structures (6(a)–6(d)), (b) in 6(a) DCHAF structure where air-holes are stretched horizontally while the doped core is kept circular for different wavelengths. The wavelength increases in a step of 5 nm from solid blue curve (1530 nm) to solid black curve (1560 nm). It is seen that the peak of the PD shifts to the left as wavelength decreases, suggesting low PMD values.
Fig. 10. Mean PMD coefficient for 6(a) DCHAF structure as a function of wavelength.

5. Summary

To conclude our work, we have proposed a very simple and novel design of a broadband DCHAF. The designed DCHAF structure whose geometrical parameters were optimized by GA, demonstrates a large negative dispersion of -550 ps/nm/km at 1550 nm that is almost two to five times larger in magnitude than the conventional broadband DCFs with nearly matched RDS. We have also carried out the impact of fiber tolerances on the dispersion characteristics. It was noticed that a ±2% variation in fiber parameters may change the dispersion values by ±12% with keeping matched the RDS. The proposed DCHAF also shows the capabilities to act as in-line Raman amplifier. The Raman amplification properties of DCHAF module were investigated by pumping the module in backward direction. The DCHAF module yield a peak gain of 4.2 dB with ±0.8 dB gain ripples over C-band. The DCHAF Raman amplifier exhibits noise figure of 9 dB, OSNR of 42.3 dB, and double Rayleigh backscattering of -57 dB on average.

As an importance of our proposed device in optical fiber links, its PMD characteristics have been computed numerically by considering a certain deformation either in air-holes or in the core or both at a same time. It was revealed from numerical simulations that the distortion in air-holes leads to higher birefringence and thus larger PMD values. A PMD of 11.36 ps/√km was obtained at 1550 nm wavelength for a DCHAF whose air-holes are distorted and are oriented horizontally when a large length of DCHAF has been selected. We look forward for the fabrication of the proposed DCHAF structure.

Acknowledgements

Authors acknowledge to the Global COE (Centers of Excellence) Program “Center for Next-Generation Information Technology based on Knowledge Discovery and Knowledge Federation”. Shailendra K. Varshney kindly acknowledges to the Japan Society for Promotion of Science (JSPS) for their support in carrying out this work.

References and Links

1.

L. Gruner-Nielsen, M. Wandel, P. Kristensen, C. Jorgensen, L.U. Jorgensen, B. Edvold, B. Palsdottir, and D. Jakobsen, “Dispersion-compensating fibers,” J. Lightwave Technol. 23, 3566–3579 (2005). [CrossRef]

2.

K. Thyagarajan, R.K. Varshney, P. Palai, A.K. Ghatak, and I.C. Goyal, “A novel design of a dispersion compensating fiber,” IEEE Photon. Technol. Lett. 8, 1510–1512 (1996). [CrossRef]

3.

J.-L. Auguste, R. Jindal, J.-M. Blondy, M. Clapeau, J. Marcou, B. Dussardier, G. Monnom, D.B. Ostrowsky, B.P. Pal, and K. Thyagarajan, “-1800 ps/(nm.km) chromatic dispersion of 1.55 µm in dual concentric core fibre,” Electron. Lett. 36, 1689–1691 (2000). [CrossRef]

4.

P.St.J. Russell, “Photonic crystal fiber,” Science 288, 358–362 (2003). [CrossRef]

5.

J.C. Knight, “Photonic crystal fibers and fiber lasers,” J. Opt. Soc. Am. B 24, 1661–1668 (2007). [CrossRef]

6.

A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibres (Kluwer Academic, The Netherlands, 2003). [CrossRef]

7.

T.A. Birks, J.C. Knight, and P.St.J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett.22, 961–963 (1997).

8.

K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fibers: Application to ultra-flattened dispersion,” Opt. Express 11, 843–852 (2003). [CrossRef] [PubMed]

9.

S.K. Varshney, K. Saitoh, and M. Koshiba, “A novel design for dispersion-compensating photonic crystal fiber Raman amplifier,” IEEE Photon. Technol. Lett. 17, 2062–2064 (2005). [CrossRef]

10.

S.K. Varshney, T. Fujisawa, K. Saitoh, and M. Koshiba, “Novel design of inherently gain-flattened discrete highly nonlinear photonic crystal fiber Raman amplifier and dispersion compensation using a single pump in C-band,” Opt. Express 13, 9516–9526 (2005). [CrossRef] [PubMed]

11.

S.K. Varshney, T. Fujisawa, K. Saitoh, and M. Koshiba, “Design and analysis of a broadband dispersion compensating photonic crystal fiber Raman amplifier operating in S-band,” Opt. Express 14, 3528–3540 (2006). [CrossRef] [PubMed]

12.

S.K. Varshney, K. Saitoh, M. Koshiba, and P.J. Roberts, “Analysis of a realistic and idealized dispersioncompensating photonic crystal fiber Raman amplifier,” Opt. Fiber Technol. 13, 174–179 (2007). [CrossRef]

13.

T. Fujisawa, K. Saitoh, K. Wada, and M. Koshiba, “Chromatic dispersion profile optimization of dualconcentric-core photonic crystal fibers for broadband dispersion compensation,” Opt. Express 14, 893–900 (2006). [CrossRef] [PubMed]

14.

F. Gérôme, J.-L. Auguste, and J.-M. Blondy, “Design of dispersion-compensating fibers based on a dual-concentric-core photonic crystal fiber,” Opt. Lett. 29, 2725–2727 (2005). [CrossRef]

15.

B.J. Mangan, F. Couny, L. Farr, A. Langford, P.J. Roberts, D.P. Williams, M. Banham, M.W. Mason, D.F. Murphy, E.A.M. Brown, H. Sabert, T.A. Birks, J.C. Knight, and P.St.J. Russell, “Slope-matched dispersion-compensating photonic crystal fibre,” in Proceedings of Conference on Lasers and Electro-Optics (CLEO 2004), paper CPDD3, San Francisco, CA, (2004). [PubMed]

16.

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss,” Opt. Express 9, 681–686 (2001). [CrossRef] [PubMed]

17.

K. Saitoh, Y. Tsuchida, and M. Koshiba, “Bending-insensitive single-mode hole-assisted fibers with reduced splice loss,” Opt. Lett. 30, 1779–1781 (2005). [CrossRef] [PubMed]

18.

B.L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Technol. Lett. 4, 1066–1069 (1992). [CrossRef]

19.

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,” IEEE J. Quantum Electron. 33, 927–933 (2002). [CrossRef]

20.

K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express 14, 11128–11141 (2006). [CrossRef] [PubMed]

21.

M. Bottacini, F. Poli, A. Cucinotta, and S. Selleri, “Modeling of photonic crystal fiber Raman amplifiers,” J. Lightwave Technol. 22, 1707–1713 (2004). [CrossRef]

22.

M. Onishi, Y. Koyano, M. Shigematsu, H. Kanamori, and H. Nishimura, “Dispersion compensating fiber with a high figure of merit of 250 ps/nm/dB,” Electron. Lett. 30, 161–163 (1994). [CrossRef]

23.

Through e-mail correspondence with crystal-fiber company (www.crystal-fiber.com).

24.

G. Millot, A. Sauter, J.M. Dudley, L. Provino, and R.S. Windeler, “Polarization mode dispersion and vectorial modulational instability in air-silica microstructure fiber,” Opt. Lett. 27, 695–697 (2002). [CrossRef]

25.

A.O. Dal Forno, A. Paradisi, R. Passy, and J.P. von der Weid, “Experimental and theoretical modeling of polarization-mode dispersion in single-mode fibers,” IEEE Photon. Technol. Lett. 12, 296–298 (2000). [CrossRef]

26.

D.A. Nolan, X. Chen, and M. Li, “Fibers with low polarization-mode dispersion,” J. Lightwave Technol. 22, 1066–1077 (2004). [CrossRef]

27.

D. Gupta, A. Kumar, and K. Thyagarajan, “Polarization mode dispersion in single mode optical fibers due to core-ellipticity,” Opt. Commun. 263, 36–41 (2006). [CrossRef]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2400) Fiber optics and optical communications : Fiber properties
(060.4005) Fiber optics and optical communications : Microstructured fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: October 22, 2007
Revised Manuscript: December 10, 2007
Manuscript Accepted: December 10, 2007
Published: December 12, 2007

Citation
Kunimasa Saitoh, Shailendra K. Varshney, and Masanori Koshiba, "Dispersion, birefringence, and amplification characteristics of newly designed dispersion compensating hole-assisted fibers," Opt. Express 15, 17724-17735 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-26-17724


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References

  1. L. Gruner-Nielsen, M. Wandel, P. Kristensen, C. Jorgensen, L.U. Jorgensen, B. Edvold, B. Palsdottir, and D. Jakobsen, "Dispersion-compensating fibers," J. Lightwave Technol. 23, 3566-3579 (2005). [CrossRef]
  2. K. Thyagarajan, R.K. Varshney, P. Palai, A.K. Ghatak, and I.C. Goyal, "A novel design of a dispersion compensating fiber," IEEE Photon. Technol. Lett. 8, 1510-1512 (1996). [CrossRef]
  3. J.-L. Auguste, R. Jindal, J.-M. Blondy, M. Clapeau, J. Marcou, B. Dussardier, G. Monnom, D.B. Ostrowsky, B.P. Pal, and K. Thyagarajan, "?1800 ps/(nm.km) chromatic dispersion of 1.55 ?m in dual concentric core fibre," Electron. Lett. 36, 1689-1691 (2000). [CrossRef]
  4. P.St.J. Russell, "Photonic crystal fiber," Science 288, 358-362 (2003). [CrossRef]
  5. J.C. Knight, "Photonic crystal fibers and fiber lasers," J. Opt. Soc. Am. B 24, 1661-1668 (2007). [CrossRef]
  6. A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibres (Kluwer Academic, The Netherlands, 2003). [CrossRef]
  7. T.A. Birks, J.C. Knight, and P.St.J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
  8. K. Saitoh, M. Koshiba, T. Hasegawa, E. Sasaoka, "Chromatic dispersion control in photonic crystal fibers: Application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003). [CrossRef] [PubMed]
  9. S.K. Varshney, K. Saitoh, and M. Koshiba, "A novel design for dispersion-compensating photonic crystal fiber Raman amplifier," IEEE Photon. Technol. Lett. 17, 2062-2064 (2005). [CrossRef]
  10. S.K. Varshney, T. Fujisawa, K. Saitoh, and M. Koshiba, "Novel design of inherently gain-flattened discrete highly nonlinear photonic crystal fiber Raman amplifier and dispersion compensation using a single pump in C-band," Opt. Express 13, 9516-9526 (2005). [CrossRef] [PubMed]
  11. S.K. Varshney, T. Fujisawa, K. Saitoh, and M. Koshiba, "Design and analysis of a broadband dispersion compensating photonic crystal fiber Raman amplifier operating in S-band," Opt. Express 14,3528-3540 (2006). [CrossRef] [PubMed]
  12. S.K. Varshney, K. Saitoh, M. Koshiba, and P.J. Roberts, "Analysis of a realistic and idealized dispersion- compensating photonic crystal fiber Raman amplifier," Opt. Fiber Technol. 13, 174-179 (2007). [CrossRef]
  13. T. Fujisawa, K. Saitoh, K. Wada, and M. Koshiba, "Chromatic dispersion profile optimization of dual-concentric-core photonic crystal fibers for broadband dispersion compensation," Opt. Express 14, 893-900 (2006). [CrossRef] [PubMed]
  14. F. Gérôme, J.-L. Auguste, and J.-M. Blondy, "Design of dispersion-compensating fibers based on a dual-concentric-core photonic crystal fiber," Opt. Lett. 29, 2725-2727 (2005). [CrossRef]
  15. B.J. Mangan, F. Couny, L. Farr, A. Langford, P.J. Roberts, D.P. Williams, M. Banham, M.W. Mason, D.F. Murphy, E.A.M. Brown, H. Sabert, T.A. Birks, J.C. Knight, and P.St.J. Russell, "Slope-matched dispersion-compensating photonic crystal fibre," in Proceedings of Conference on Lasers and Electro-Optics (CLEO 2004), paper CPDD3, San Francisco, CA, (2004). [PubMed]
  16. T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, "Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss," Opt. Express 9, 681-686 (2001). [CrossRef] [PubMed]
  17. K. Saitoh, Y. Tsuchida, and M. Koshiba, "Bending-insensitive single-mode hole-assisted fibers with reduced splice loss," Opt. Lett. 30, 1779-1781 (2005). [CrossRef] [PubMed]
  18. B.L. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1069 (1992). [CrossRef]
  19. K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers," IEEE J. Quantum Electron. 33, 927-933 (2002). [CrossRef]
  20. K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, "Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends," Opt. Express 14, 11128-11141 (2006). [CrossRef] [PubMed]
  21. M. Bottacini, F. Poli, A. Cucinotta, and S. Selleri, "Modeling of photonic crystal fiber Raman amplifiers," J. Lightwave Technol. 22, 1707-1713 (2004). [CrossRef]
  22. M. Onishi, Y. Koyano, M. Shigematsu, H. Kanamori, and H. Nishimura, "Dispersion compensating fiber with a high figure of merit of 250 ps/nm/dB," Electron. Lett. 30, 161-163 (1994). [CrossRef]
  23. Through e-mail correspondence with crystal-fiber company (www.crystal-fiber.com).
  24. G. Millot, A. Sauter, J.M. Dudley, L. Provino, and R.S. Windeler, "Polarization mode dispersion and vectorial modulational instability in air-silica microstructure fiber," Opt. Lett. 27, 695-697 (2002). [CrossRef]
  25. A.O. Dal Forno, A. Paradisi, R. Passy, and J.P. von der Weid, "Experimental and theoretical modeling of polarization-mode dispersion in single-mode fibers," IEEE Photon. Technol. Lett. 12, 296-298 (2000). [CrossRef]
  26. D.A. Nolan, X. Chen, and M. Li, "Fibers with low polarization-mode dispersion," J. Lightwave Technol. 22, 1066-1077 (2004). [CrossRef]
  27. D. Gupta, A. Kumar, and K. Thyagarajan, "Polarization mode dispersion in single mode optical fibers due to core-ellipticity," Opt. Commun. 263, 36-41 (2006). [CrossRef]

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