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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 21 — Oct. 17, 2007
  • pp: 13547–13556
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Experimental investigations of bending loss oscillations in large mode area photonic crystal fibers

T. Martynkien, J. Olszewski, M. Szpulak, G. Golojuch, W. Urbanczyk, T. Nasilowski, F. Berghmans, and H. Thienpont  »View Author Affiliations


Optics Express, Vol. 15, Issue 21, pp. 13547-13556 (2007)
http://dx.doi.org/10.1364/OE.15.013547


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Abstract

We demonstrate experimentally that bending loss in large mode area photonic crystal fibers oscillates with wavelength. To do so we carried out loss measurements for different fiber bend radii and for different angular orientations. These results confirm the oscillatory behavior of bending loss vs. wavelength as predicted recently by numerical analysis [J. Olszewski et al., Opt. Express 13, 6015 (2005)]. We also found good agreement between our measurement results and our simulations relying on a finite element method with perfectly matched layers and an equivalent index model.

© 2007 Optical Society of America

1. Introduction

The impact of angular orientation on bending induced loss was investigated for the first time in [7

7. J. C. Baggett, T. M. Monro, K. Furusawa, V. Finazzi, and D.J. Richardson. “Understanding bending losses in holey optical fibers,” Opt. Commun. 227, 317–335 (2003). [CrossRef]

]. However, due to the fact that the core of the analyzed PCFs was offset from the fiber symmetry axes (the extent of the cladding varied from seven to eight layers of holes around the core, depending on the angular direction), it was impossible to correlate the measurement results with the hexagonal symmetry of the micro-structured cladding.

In this contribution, we present the results of a systematic measurement campaign of bending loss in large mode area PCF (LMA-20, Λ = 13.2 μm, d/Λ = 0.485). To evidence the oscillations we measured the loss in the spectral range of 0.8–1.6 μm as a function of bending radius and for different angular orientations of the fiber with respect to the bending plane. Our results show that both the depth and the period of loss oscillations depend significantly on the angular orientation of the fiber. This is also confirmed by our numerical calculations relying on the Finite Element Method (FEM) with anisotropic perfectly matched layers (PML) boundary conditions.

2. Theoretical background

To analyze the effect of bending loss in the PCFs one usually considers an equivalent step-index model [2–6

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

]. According to this model the fiber consists of two regions: the core and the holey cladding extending to infinity. Bending loss estimated using this approach changes monotonically with wavelength (λ), and bend radius (R). In a real fiber however the holey cladding has limited size and therefore the cross section of the typical index-guided PCF comprises three regions: the core, the holey part of the cladding surrounding the core, and the outer part of the cladding made of solid silica. One can therefore state that a practical PCF is more equivalent to a conventional double-clad fiber. It is well known [11–15

11. A. J. Harris and P.F. Castle, “Bend loss measurement on high numerical aperture single-mode fibers as function of wavelength and bend radius,” J. Lightwave Technol. 4, 34–40 (1986). [CrossRef]

] that bending loss in conventional double-clad fibers exhibits oscillations superimposed on the monotonic loss increase with λ and 1/R. As we already demonstrated in earlier work a similar mechanism is responsible for bending loss oscillations in PCFs [9

9. J. Olszewski, M. Szpulak, and W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13, 6015 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6015. [CrossRef] [PubMed]

, 10

10. J. Olszewski, M. Szpulak, T. Martynkien, W. Urbanczyk, F. Berghmans, T. Nasilowski, and H. Thienpont, “Analytical evaluation of bending loss oscillations in photonic crystal fibers,” Opt. Commun. 269, 261–270 (2007). [CrossRef]

].

Fig. 1. Spectral dependence of bending loss (a) for different angular orientations of the fiber with respect to the bending plane (b). Numbers associated to selected loss peaks correspond to the order of the phase matched cladding modes shown in Fig. 2. Calculations were carried out for a bending radius R = 80 mm.

The calculations of bending loss for the purposes of this and our earlier work were carried out using FEM with PML boundary conditions [16

16. Y. Tsuji and M. Koshiba, “Complex modal analysis of curved optical waveguides using a full-vectorial finite element method with perfectly matched layer boundary conditions,” Electromagnetics 24, 39–48 (2004). [CrossRef]

]. The curved fiber was replaced by a straight fiber with an equivalent refractive index distribution [9

9. J. Olszewski, M. Szpulak, and W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13, 6015 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6015. [CrossRef] [PubMed]

, 10

10. J. Olszewski, M. Szpulak, T. Martynkien, W. Urbanczyk, F. Berghmans, T. Nasilowski, and H. Thienpont, “Analytical evaluation of bending loss oscillations in photonic crystal fibers,” Opt. Commun. 269, 261–270 (2007). [CrossRef]

]:

neqxy=nxyexp(pR),
(1)

where p = x or y, depending on the bending direction, and R stands for the bend radius. A rectangular computational domain was surrounded by the PML composed of three regions with different complex stretching variables. The calculation domain was divided into 140 000 triangular elements and the PML parameters were chosen as described previously in [9

9. J. Olszewski, M. Szpulak, and W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13, 6015 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6015. [CrossRef] [PubMed]

].

Fig. 2. Distribution of the dominant component of the electric field in selected, y-polarized cladding modes. The calculations were performed for the angular orientations φ= 0° (a). φ= 30° (b) and φ= 15° (c) of the fiber with respect to the bending plane and for radii assuring phase matching condition of respective cladding modes with the fundamental mode, λ = 0.83 μm.

When the fiber is bent at φ= 30° or φ= 15° the leaking field is reflected from the tilted boundaries of the holey region. Then, as it is shown in Fig. 2(b, c), the cladding modes have lower symmetry and arise at larger distance from the core compared to bends at φ= 0°. It causes the overlap integral and therefore the coupling coefficients between the fundamental mode and the cladding modes to be much smaller. Hence we observe only relatively weak oscillations in the loss characteristics for bends at φ= 30° and φ= 15°.

The above arguments indicate that special care has to be taken during the experiment to control the angular orientation of the fiber in order to observe the peaks in the bending loss characteristics.

3. Bending loss measurements

We measured bending loss in commercially available large mode area PCF (LMA - 20) produced by Crystal Fibre A/S (Fig. 3(a)). The geometrical parameters of this fiber determined from a Scanning Electron Microscope picture are as follows: pitch Λ= 13.2 μm, fill factor d/Λ = 0.485, and 7 layers of holes surrounding the core.

Fig. 3. Cross-section of the investigated large mode area fiber LMA-20 (a) and scheme of the experimental set-up for bending loss measurements (b).

A schematic diagram of the set-up for loss measurements vs. bending radii for different angular orientations of the fiber is shown in Fig. 3(b). We first couple unpolarized white light from a halogen lamp into the straight fiber and record a reference transmission spectrum using an optical spectrum analyzer (OSA). We then also record the transmission spectrum for the fiber bent on a drum into a half-loop (Fig. 3(b)). The bending loss expressed in dB/half-turn is determined as the difference between the transmission spectra obtained for the two measurements.

To control the angular orientation of the fiber with respect to the bending plane during loss measurements (Fig. 1) we use fiber chucks and rotators positioned at the beginning and at the end of the half turn. This allows rotating the fiber independently at these two points. The initial angular orientation of the fiber over the full length of the half-turn is adjusted using visual inspection through a microscope. The side-views of the fiber in three different angular orientations as well as the diagrams of corresponding cross-sections are shown in Fig. 4. Only the alignment at φ= 0° can be easily recognized by microscopic inspection (Fig. 4(a)). In this particular case the air holes form vertical columns located at largest distance between each other thus allowing the light coming from the bottom to pass undistorted through the solid glass paths between the air holes. For such alignment, symmetrical and clearly visible bright lines can be observed in the microscope image, Fig. 4(b). For other orientations of the fiber, the light is either scattered by air holes located on its way therefore causing the bright lines to disappear (Fig. 4(c)) or the glass paths are too narrow to produce a high contrast fringe pattern (Fig. 4(d)).

Fig. 4. Schematic diagram illustrating the procedure of angular alignment using an inspection microscope (a). Side view and corresponding diagrams of the fiber cross-sections for different angular orientations with respect to the bending plane (dashed line) (b - d).

We estimated that the accuracy of the angular positioning using visual inspection is about ±3°. By fine tuning the orientation to that where bending induced oscillations have the highest amplitude we can align the azimuth of the bending plane at φ =0° with a precision of 1°. The uncertainty on the initial angular alignment can be further verified by taking advantage of the 6-fold rotational symmetry of the fiber, which should result in identical bending loss curves for rotations with an angle ±α around its initial position φ =0° (xz-bending plane).

Fig. 5. Bending loss measured vs. wavelength for different angular orientations of the LMA-20 fiber (a - d), bending radius R = 25 mm.

In Fig. 5, the wavelength dependence of the loss for a bend radius R =25 mm and different angular orientations (φ = ±60°, φ = ±30°, φ = ±15° and φ = ±8°) of the fiber are compared with the loss curves obtained for the xz-bending plane (φ = 0°). For longer wavelengths the registered signals were relatively low and noisy. We therefore suppressed the noise using a moving average filter with a 3 nm window width. Figure 5 shows that departure from the xz-bending plane causes an increase of the number of peaks and lowers their amplitude. On the other hand the peak locations can be very well reproduced for symmetrical rotations (see for example Fig. 5(b-c)) This confirms the accuracy of the initial angular alignment of the investigated fiber in the xz-bending plane.

The length of the straight lead-in and lead-out sections of the investigated fiber was about 1.5 m. The angular orientation of the fiber was only controlled in the bent area, therefore the leading sections could be slightly twisted. However, as already demonstrated in [17

17. J. Zhou, K. Tajima, K. Nakajima, K. Kurokawa, T. Matsui, C. Fukai, and I. Sankawa, “PMD characteristics of twisted photonic crystal fibre,” Electron. Lett. 41, 403–405 (2005). [CrossRef]

], the effect of twist in the straight LMA PCF has practically no influence on its transmission loss characteristics and therefore didn’t affect our measurement results.

Fig. 6. Bending loss measured as a function of angular orientation of the LMA-20 fiber for different wavelengths and for the same bending radius R = 25 mm.

To evaluate the effect of the angular orientation on bending loss we chose three different wavelengths (λ= 1050, 1180, 1420 nm) and a fixed bending radius R = 25 mm, while the fiber azimuth was changed from -70° to +70° in steps of 2°. As shown in Fig. 6 the 6-fold symmetry of the fiber results in reproducible loss curves with a period ∆φ= 60°. The loss variations against fiber azimuth are significant. For example, at λ= 1050 nm the difference between the loss minimum (φ= 0°) and maximum (φ= ±30°) is about 16 dB/half-loop.

4. Results

The spectral measurements of bending loss were carried out for the range of bending radii R = 20–60 mm with a step of 5 mm, Fig. 7. The results presented in Fig. 7 were obtained for the xz-bending plane. We consider those peaks with local highest amplitude as primary peaks The locations of corresponding primary peaks for different bending radii are indicated with color arrows. One can notice that a secondary short period modulation appears that is most probably caused by cladding modes appearing due to light reflection at the interface between the solid part of the cladding and the polymer coating. The primary and secondary loss oscillations may not be attributed to the measurements noise because they preserve their shape for different bending radii. The minimum loss induced by the half-turn bend saturates at about 0.1 dB/half-turn which correspond to the measurement resolution of our experimental set-up.

Fig. 7. Bending losses measured vs. wavelength for different bending radii. All characteristics are obtained for xz-bending plane. The positions of corresponding primary peaks are marked with black and blue arrows.

As displayed in Fig. 7 the loss increases with 1/R and 1/λ. Loss peaks also move towards longer wavelengths with increasing bend curvature 1/R. This is confirmed by numerical simulations of bending loss for the LMA-20 fiber using the approach described in section 2, Fig. 8. The calculations were performed for the xz-bending plane in the range of bending radii from 25 mm to 100 mm. The shift of the resonant loss peaks towards longer wavelength with increasing 1/R can be intuitively explained by using the tilted refractive index model. The change of the refractive index distribution in the bent fiber is proportional to the distance from the core and the curvature 1/R (eq. 1). Increasing the bend curvature affects mostly the cladding modes by moving them closer to the boundary between the solid and the holey part of the cladding. Consequently the effective indices of the cladding modes are increased due to bending, while the effective index of the fundamental mode remains almost unchanged. To maintain phase matching between the fundamental mode and the corresponding cladding modes one has to lower the effective index of the cladding modes. This is only possible by increasing the wavelength, which results in the displacement of the cladding modes back towards the fiber core.

Fig. 8. Bending losses calculated vs. wavelength for different bending radii.

Figure 8 also shows the increasing complexity of the features of calculated loss curves for radii smaller than 35 mm. More particularly the splitting of loss peaks and the appearance of secondary maxima on the short wavelength slope of the primary maxima.

Fig. 9. Comparison of measured (blue lines) and calculated (red lines) bending loss characteristics obtained for the xz-bending plane, R = 35 mm (a) and R = 50 mm (b). The markers P1 and P2 indicate the corresponding loss peaks in numerical and experimental characteristics.

In Fig. 9 we collate the results of measurements and numerical simulations obtained for the xz-bending plane and radii R = 35 mm and R =50 mm, respectively. There are clearly visible differences in oscillation depths and in the locations of small secondary peaks. This mismatch most probably stems from disregarding the second reflecting interface between the solid part of the cladding and the polymer coating in numerical simulations, and perhaps from the limited precision of the angular alignment of the fiber during the measurements. The corresponding primary loss peaks can nevertheless be clearly identified in experimental and numerical characteristics. Moreover, the calculated and measured shifts of these peaks versus bending radii are in very good agreement. The dependences of wavelengths λP1, λP2 defining the positions of the peaks P1 and P2 vs. fiber curvature are displayed in Fig. 10. λP1 and λP2 change linearly with 1/R. Furthermore, using linear regression we obtained almost identical slopes for the experimental and calculated dependences of λP1 and λP2 on 1/R. The difference between the linear coefficients for the experimental and calculated shifts does not exceed 1% for both peaks. This validates our theoretical model that associates the origin of the primary loss peaks for the xz-bending with the energy coupling between the fundamental mode and the cladding modes of high symmetry (i.e., with field oscillations only in horizontal direction).

Fig. 10. Comparison of experimental (blue circle and lines) and calculated (red circle and lines) wavelength shift corresponding to the P1 (a) and P2 (b) peaks vs. bending curvature 1/R.

5. Conclusion

We studied experimentally the effect of bending loss oscillations in large mode area photonic crystal fibers. Our results confirmed earlier theoretical predictions [9

9. J. Olszewski, M. Szpulak, and W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13, 6015 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6015. [CrossRef] [PubMed]

], indicating that both the amplitude and the period of these oscillations depend on the angular orientation of the fiber with respect to the bending plane. The oscillations are particularly well pronounced when the radiative component of the fundamental mode is reflected from the flat boundary of the holey cladding, i.e. for xz-bends. Therefore, to control the fiber azimuth during the loss measurements, we first aligned the fiber in the xz-bending plane by using a special procedure that involves a visual inspection of the fiber side-view along its full length. The effectiveness of the proposed alignment procedure was confirmed by the invariance of the loss characteristics obtained for the fiber rotated by ±φ around its initial position as well as by the periodicity of loss characteristics measured for three different wavelengths vs. angular orientation of the fiber in the range of φ=±70°.

Our experimental results obtained for the xz-bending prove that the locations of the primary loss peaks are shifted linearly towards longer wavelength for increasing bending curvature 1/R. We compared the wavelength shifts of selected primary peaks against 1/R for experimental and calculated loss curves. The discrepancy between calculated and measured shifts of both analyzed peaks was lower than 1%. This excellent agreement validates our theoretical model presented in [9

9. J. Olszewski, M. Szpulak, and W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13, 6015 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6015. [CrossRef] [PubMed]

], which connects the origin of the loss peaks with energy coupling between the fundamental mode and the cladding modes of respective symmetry arising due to field reflection from the boundary between the solid and holey part of the cladding.

The research presented in this paper also opens perspectives for applications, for example in the field of spectrally encoded fiber bending or deformation sensors.

Acknowledgments

This work was partially supported by the EC 6th FP Network of Excellence on Micro-Optics “NEMO” and by the COST 299 Action. In addition, T. M. acknowledges funding from the FWO under Contract No. GP02906N-FWOSL22.

References and links

1.

P. St.J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24, 4729–4749 (2006). [CrossRef]

2.

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

3.

T. Sørensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. B. Libori, “Macro-bending loss properties of photonic crystal fibre,” Electron. Lett. 37, 287–289 (2001). [CrossRef]

4.

T. Sørensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, “Spectral macro-bending loss considerations for photonic crystal fibres,” IEE Proc.-Optoelectron. 149, 206–210 (2002). [CrossRef]

5.

M.D. Nielsen, J.R. Folkenberg, N.A. Mortensen, and A. Bjarklev, “Bandwidth comparison of photonic crystal fibers and conventional single-mode fibers,” Opt. Express 12, 430 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-3-430. [CrossRef] [PubMed]

6.

M. D. Nielsen, N. A. Mortensen, M. Albertsen, J. R. Folkenberg, A. Bjarklev, and D. Bonacinni “Predicting macrobending loss for large-mode area photonic crystal fibers,” Opt. Express 12, 1775 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1775. [CrossRef] [PubMed]

7.

J. C. Baggett, T. M. Monro, K. Furusawa, V. Finazzi, and D.J. Richardson. “Understanding bending losses in holey optical fibers,” Opt. Commun. 227, 317–335 (2003). [CrossRef]

8.

Y. Tsuchida, K. Saitoh, and M. Koshiba “Design and characterisation of single-mode holey fibers with low bending losses,” Opt. Express 13, 4770 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4770. [CrossRef] [PubMed]

9.

J. Olszewski, M. Szpulak, and W. Urbanczyk, “Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers,” Opt. Express 13, 6015 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6015. [CrossRef] [PubMed]

10.

J. Olszewski, M. Szpulak, T. Martynkien, W. Urbanczyk, F. Berghmans, T. Nasilowski, and H. Thienpont, “Analytical evaluation of bending loss oscillations in photonic crystal fibers,” Opt. Commun. 269, 261–270 (2007). [CrossRef]

11.

A. J. Harris and P.F. Castle, “Bend loss measurement on high numerical aperture single-mode fibers as function of wavelength and bend radius,” J. Lightwave Technol. 4, 34–40 (1986). [CrossRef]

12.

R. Morgan, J.S. Barton, P.G. Harper, and J.D.C. Jones, “Wavelength dependence of bending loss in mononmode optical fibers: effect of the fiber buffer coating,” Opt. Lett. 15, 947–949 (1990). [CrossRef] [PubMed]

13.

H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. 10, 544–551 (1992). [CrossRef]

14.

L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]

15.

Q. Wang, G. Farrell, and T. Freir, “Theoretical and experimental investigations of macro-bend losses for standard single mode fibers,” Opt. Express 13, 4476 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4476. [CrossRef] [PubMed]

16.

Y. Tsuji and M. Koshiba, “Complex modal analysis of curved optical waveguides using a full-vectorial finite element method with perfectly matched layer boundary conditions,” Electromagnetics 24, 39–48 (2004). [CrossRef]

17.

J. Zhou, K. Tajima, K. Nakajima, K. Kurokawa, T. Matsui, C. Fukai, and I. Sankawa, “PMD characteristics of twisted photonic crystal fibre,” Electron. Lett. 41, 403–405 (2005). [CrossRef]

OCIS Codes
(060.2300) Fiber optics and optical communications : Fiber measurements
(060.2400) Fiber optics and optical communications : Fiber properties
(230.3990) Optical devices : Micro-optical devices

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: June 4, 2007
Revised Manuscript: August 20, 2007
Manuscript Accepted: September 24, 2007
Published: October 1, 2007

Citation
T. Martynkien, J. Olszewski, M. Szpulak, G. Golojuch, W. Urbanczyk, T. Nasilowski, F. Berghmans, and H. Thienpont, "Experimental investigations of bending loss oscillations in large mode area photonic crystal fibers," Opt. Express 15, 13547-13556 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-13547


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References

  1. P. St.J. Russell, "Photonic-Crystal Fibers," J. Lightwave Technol. 24,4729-4749 (2006). [CrossRef]
  2. T. A. Birks, J. C. Knight, P. St.J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22,961-963 (1997). [CrossRef] [PubMed]
  3. T. Sørensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. B. Libori, "Macro-bending loss properties of photonic crystal fibre," Electron. Lett. 37,287-289 (2001). [CrossRef]
  4. T. Sørensen, J. Broeng, A. Bjarklev, T. P. Hansen, E. Knudsen, S. E. B. Libori, H. R. Simonsen, and J. R. Jensen, "Spectral macro-bending loss considerations for photonic crystal fibres," IEE Proc.-Optoelectron. 149,206-210 (2002). [CrossRef]
  5. M.D. Nielsen, J.R. Folkenberg, N.A. Mortensen, and A. Bjarklev, "Bandwidth comparison of photonic crystal fibers and conventional single-mode fibers," Opt. Express 12,430 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-3-430. [CrossRef] [PubMed]
  6. M. D. Nielsen, N. A. Mortensen, M. Albertsen, J. R. Folkenberg, A. Bjarklev, and D. Bonacinni "Predicting macrobending loss for large-mode area photonic crystal fibers," Opt. Express 12,1775 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-8-1775. [CrossRef] [PubMed]
  7. J. C. Baggett, T. M. Monro, K. Furusawa,V. Finazzi, D.J. Richardson. "Understanding bending losses in holey optical fibers," Opt. Commun. 227,317-335 (2003). [CrossRef]
  8. Y. Tsuchida, K. Saitoh, and M. Koshiba "Design and characterisation of single-mode holey fibers with low bending losses," Opt. Express 13,4770 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4770. [CrossRef] [PubMed]
  9. J. Olszewski, M. Szpulak, and W. Urbanczyk, "Effect of coupling between fundamental and cladding modes on bending losses in photonic crystal fibers," Opt. Express 13,6015 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6015. [CrossRef] [PubMed]
  10. J. Olszewski, M. Szpulak, T. Martynkien, W. Urbanczyk, F. Berghmans, T. Nasilowski, H. Thienpont, "Analytical evaluation of bending loss oscillations in photonic crystal fibers," Opt. Commun. 269,261-270 (2007). [CrossRef]
  11. A. J. Harris and P.F. Castle, "Bend loss measurement on high numerical aperture single-mode fibers as function of wavelength and bend radius," J. Lightwave Technol. 4,34-40 (1986). [CrossRef]
  12. R. Morgan, J.S. Barton, P.G. Harper and J.D.C. Jones, "Wavelength dependence of bending loss in mononmode optical fibers: effect of the fiber buffer coating," Opt. Lett. 15,947-949 (1990). [CrossRef] [PubMed]
  13. H. Renner, "Bending losses of coated single-mode fibers: a simple approach," J. Lightwave Technol. 10,544-551 (1992). [CrossRef]
  14. L. Faustini and G. Martini, "Bend loss in single-mode fibers," J. Lightwave Technol. 15,671-679 (1997). [CrossRef]
  15. Q. Wang, G. Farrell, T. Freir, "Theoretical and experimental investigations of macro-bend losses for standard single mode fibers," Opt. Express 13,4476 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-12-4476. [CrossRef] [PubMed]
  16. Y. Tsuji, M. Koshiba, "Complex modal analysis of curved optical waveguides using a full-vectorial finite element method with perfectly matched layer boundary conditions," Electromagnetics 24,39-48 (2004). [CrossRef]
  17. J. Zhou, K. Tajima, K. Nakajima, K. Kurokawa, T. Matsui, C. Fukai, I. Sankawa, "PMD characteristics of twisted photonic crystal fibre," Electron. Lett. 41,403 - 405 (2005). [CrossRef]

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