FDTD分析光子晶体光纤弯曲

Bending loss analyses of photonic crystalfibers based on thefinite-difference time-domain methodNgoc Hai Vu,1In-Kag Hwang,1,*and Yong-Hee Lee21Department of Physics,Chonnam National University,300Yongbong-dong,Buk-gu,Gwangju500-757,South Korea 2Department of Physics,Korea Advanced Institute of Science and Technology,373-1Guseong-dong,Yuseong-gu,Daejeon305-701,South Korea*Corresponding author:ikhwang@chonnam.ac.krReceived October8,2007;revised November29,2007;accepted December6,2007;posted December11,2007(Doc.ID88351);published January8,2008This is a report on an effective simulation method for the bending loss analyses of photonic crystalfibers. This method is based on the two-dimensionalfinite-difference time-domain algorithm and a conformal trans-formation of the refractive index profile.We observed the temporal dynamics of light waves in a bentfiber in a simulation and obtained the bending loss as a function of bend radius and optical wavelength for the com-mercial photonic crystalfibers.The accuracy of this method was verified by good agreement between the simulation and experimental data.©2008Optical Society of AmericaOCIS codes:060.5295,060.2280,060.2310.First proposed in1995,photonic crystalfibers (PCFs),with silica–air microstructures,attract many researchers as thesefibers have unique applications in ultrawide-band transmission,supercontinuum generation,high power delivery,optical amplifiers, and other functional devices[1].One of the important issues regarding the practical development of these PCFs concerns their bending loss properties.When a fiber is bent,the modalfield distorts outwards in the direction of the bend and a radiation loss occurs.The bending loss is usually regarded as an adverse effect in the context of optical transmission.However,the bentfibers can also be used as a new and unique op-tical component employable in optical communica-tions or optical sensing[2].It is important to be able to accurately estimate the bending loss of a givenfi-ber structure for the design and characterization of various PCFs.The complicated microstructure in a PCF makes the calculation a challenging problem.Most of the analytical methods such as antenna theory developed for conventionalfibers with circularly symmetric in-dex profiles cannot be directly applied to PCFs.Here, we,for thefirst time to our knowledge,adopted a two-dimensionalfinite-difference time-domain(2D-FDTD)algorithm[3]for the simulation of optical propagation in a bent PCF.The three-dimensional (3D)structure of the bentfiber was transformed into a two-dimensional straightfiber by using conformal mapping of the refractive index profile of the PCF. The temporal evolution of the opticalfield was prop-erly interpreted to yield the bending loss per unit length.We compared the simulation results with the experimental results to validate the accuracy of our method.The FDTD method has some distinct features com-pared with other previous methods such as the effi-cient modal model[4,5]or thefinite-element method [6]used for the calculation of bending loss of the PCF. The FDTD algorithm is a very general tool and is ap-plicable to a wide range of electromagnetic problems. It directly solves Maxwell’s equations with minimalassumptions and approximations and thus provides fairly reliable results as long as the spatial and tem-poral resolution are high enough.Recent advance-ment of computer technology allows the use of high spatial and temporal resolution,making this tech-nique more useful and popular.This method enables full access to electromagnetic waves at an arbitrary time and position,so that one can collect any desired information from these waves.Therefore it is distin-guishable from other numerical methods that provide specific information only.The direct simulation of optical propagation in bentfiber may be performed by the FDTD method ina complete3D structure offiber loops with an inputof an optical wave from one end of thefiber loops[2].In this case,we record the optical powers at several locations of thefiber loops to obtain the bending loss as a function of the propagation length.However,this approach requires huge memory sizes and long com-putation time,which is practically unacceptable.Here,we implemented a time-domain approach for the calculation of the bending loss,instead of the space-domain approach,for computation efficiency.First we imagined an infinitely longfiber with a bendradius of R b as shown in Fig.1(a).At t=0,thefiber is filled with an optical wave with a propagation con-stant␤and a uniform intensity along the length.Af-ter t=0,the optical wave starts to attenuate with arate of␣Јdue to the bending loss.Note that the op-tical intensity is always uniform over the wholefiber length during the attenuation.Finally the loss coeffi-cient per unit time͑␣Ј͒is converted to a loss coeffi-cient per unit length͑␣͒using the equation␣=␣Ј/v, where v is the velocity of light in thefiber.Since the optical wave has no variation along the length except the phase in the above situation,the computation structure for the FDTD method can be reduced to one slice of the bentfiber with an arbi-trarily small thickness͑⌬z͒as shown in Fig.1(b).Then,the3D structure of the bent piece is again sim-plified to aflat one[Fig.1(c)]by employing an equiva-lent index profile given byJanuary15,2008/Vol.33,No.2/OPTICS LETTERS1190146-9592/08/020119-3/$15.00©2008Optical Society of American eq 2͑x ,y ͒=n 2͑x ,y ͒ͩ1+2x R bͪ,where R b is the radius of curvature and n ͑x ,y ͒is therefractive index profile of the straight fiber [6–8].The bottom part of Fig.1(c)shows the transformed refrac-tive index profile of a PCF with a bend radius of 5.5mm.It clearly shows that the transformation su-perimposes a gradient onto the refractive index of the straight fiber in the direction of the bend.Therefore the final computation structure is effectively a 2D one containing only one grid along the z .The sizes of the computation grids were set to ⌬x =⌬y =⌳/20and ⌬z =⌳/100,where ⌬is the hole pitch.Those param-eters were optimized to maximize the numerical ac-curacy within a reasonable computation time.Figure 1(d)shows the recorded optical intensity E 2at the center of the fiber as a function of time.The optical intensity corresponds to I ͑t ͒=͗E 2͑t ͒͘.Here we obtained the loss factor per unit time,␣Ј,from curve fitting with the function I ͑t ͒=I 0exp ͑−␣Јt ͒.The veloc-ity of the light was calculated from v =␻/␤,where ␻was the oscillation frequency of Fig.1(d),to get the loss factor per unit length,␣=␣Ј/v .The simulation was performed for the ESM-5PCF (hole pitch,⌳=8␮m;normalized hole diameter,d /⌳=0.46)from Crystal Fibre A/S.The parameters ⌳and d were extracted directly from a scanning electron microscope image of the real fiber.The refractive in-dex of the silica was set as 1.444,and no material dispersion was assumed.Figure 2(a)shows the inten-sity distributions of the fundamental mode at ␭=1550nm for the bend in the x direction of the radiusof 3.5mm.It clearly shows that the mode of the bent fiber was asymmetric in shape and shifted towards the outside of the bend [4].We generated multiple in-tensity profiles at successive time frames in one pe-riod of the optical oscillation to observe the dynamics of the optical field,which are shown in Fig.2(b).Here we could observe a “propagating”field radiated from the center toward the outside of the bend while the “stationary”core mode was blinking at its optical fre-quency.This energy propagation across the fiber was the cause of the bending loss and resulted in the dis-sipation of the optical power in the core mode.The di-rect observation of this phenomenon could not be made by other calculation methods.We calculated the bending losses of the ESM-5PCF for several different bending radii.The plots are denoted with triangles in Fig.3(a).The typical com-putation time was about 2–3h for each data point in our case,although it depends largely on the spatial and temporal resolutions of FDTD.For comparison,the bending losses were experimentally measured us-ing a narrow-linewidth laser and an optical power-meter.Two turns of fiber loops were made for a bend radius in the range of 3.0to 7.0mm in incrementsofFig. 1.(Color online)Illustration of the simulation scheme:(a)infinitely-long bent fiber model;(b)one sliced piece of the bent fiber and its index profile along x ;(c)same as (b)after conformal transformation of the index profile;(d)E 2at the center of the fiber as a function of time,show-ing the attenuation of opticalintensity.Fig.2.(Color online)(a)Optical intensity distribution in the cross section of a bent fiber with a radius of 5.5mm (log scale)and (b)central regions of intensity profiles taken at successive times ͑⌬t =1fs ͒in one period ofoscillation.Fig.3.(Color online)Dependence of bending loss on bend-ing radius for (a)ESM-5PCF and (b)LMA-8PCF.The squares and triangles denote the experimental and simula-tion data,respectively.120OPTICS LETTERS /Vol.33,No.2/January 15,20080.3mm.For a bend radius smaller than 3.0mm,the bent fiber was easily broken;while for a radius larger than 7mm,the loss was too low for reliable and re-peatable measurements.Each measurement was re-peated three times,and its average value and devia-tion are shown with squares and error bars,respectively,in Fig.3(a).A comparison was carried out for another PCF,LMA-8(⌳=5.6␮m,d /⌳=0.49)also from Crystal Fibre A/S.The results are shown in Fig.3(b).There was reasonably good agreement be-tween the simulation and experimental results for both fibers.It is interesting to see the small bumps in the experimental curves at the bending radii of ϳ7.8mm for ESM-5PCF and ϳ4.2mm for LMA-8,which did not appear in the simulation results.These loss peaks seemed to come from resonant coupling from the core mode to a cladding mode in the multilayer structure of the cladding.Note that the full cladding structure of PCF should be included in the computation structure to investigate this phe-nomenon [9].A detailed study of its origin is under-way.The wavelength dependence of the bending loss was also calculated and measured in this report for ESM-5.In FDTD,the optical wavelength was rede-termined by simultaneous changes of the propaga-tion constant,␤,and the optical frequency,␻.For the experiment,we used an optical spectrum analyzer and a superluminescent diode with a bandwidth of Ͼ50nm.The simulation and experimental data are shown in Fig.4as dashed and solid curves,respec-tively.The strong wavelength dependence was ob-served for a small bending radius,while it was rela-tively flat for a large bending radius.The fine structures in the experimental data seemed to result from the reflection of the radiated light at thecladding–jacket or jacket–air boundaries back to the core mode.Note that the exceptionally large bending loss at R ϳ7.7mm in Fig.3(a)was observed again in Fig.4.We also found good agreement between the simulation and experiment,which again verified the accuracy of our method.We proposed an efficient numerical method for bending loss analyses of PCF,based on the 2D-FDTD method and conformal transformation of the index profile.The time-domain simulation of the optical propagation in a bent fiber provided a view of the temporal dynamics of the optical field as well as the mode profile,dispersion and the bending loss.The ac-curacy of the method was verified by good agreement between a simulation and experimental data.The technique outlined here is directly applicable to not only PCFs but also any kind of waveguides with ar-bitrary index profiles.It is important to note that this FDTD method can be easily extended by adding new functions to include nonlinear or strain effects in the simulation,which may not be available in other methods.We believe it is a useful tool for analyses and design of various microstructured fibers [10–12].This work was supported by IT R&D program of Ministry of Information and Communication and In-stitute for Information Technology Advancement (2005-S099-03,Development of photonic crystal fi-bers and their application technology for high-speed optical communication 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