Numerical algorithm for the analysis of linear and nonlinear microstructure fibres

Transcription

1 Numerical algorithm for the anali of linear and nonlinear microtructure fibre Mariuz Zdanowicz *, Marian Marciniak, Marek Jaworki, Igor A. Goncharenko National Intitute of Telecommunication, Department of Tranmiion and Fibre Optic Szachowa Str., Waraw, Poland Intitute of Electronic National Academ of Science, Logoik Trakt, 0090 Mink, Belaru ABSTRACT Thi report hortl decribe calculation method related to guiding propertie of the microtructure fibre. For the modelling of wave propagation in uch fibre in linear and nonlinear regime the finite-difference vectorial-beampropagation method (FD-VBMP) ha been choen. Thi method offer high accurac and allow invetigating longitudinall varing tructure or propagation of optical wave with amplitude varing under the effect of nonlinearit and lo. In order to take into account the effect related to the finite fibre dimenion the tranparent boundar condition (TBC) wa ued. The Split-Step Local Error method i implemented for correct etimation of the influence of nonlinearit on the wave propagation in microtructure fibre. On the bai of thi method the complete algorithm for the numerical imulation of the wave propagation in microtructure fibre under the effect of nonlinearit ha been developed. The method allow optimiing the tructure of photonic crtal fibre (PCF) for maimiing or minimiing nonlinear effect. Invetigation of the propertie of a wavelength converter baed on a microtructure fibre wa carried out uing imulation reult in the nonlinear cae. Modification of the method for optimiation of connection between microtructure fibre and conventional fibre i alo conidered. Keword: BPM, PCF, Split-Step, Local Error. INTRODUCTION.. Baic theor Developed program i baed on the ue of Finite Difference (FD) cheme in Beam Propagation Method (BPM) in frequenc regime. It allow imulating the light beam propagation in microtructure fibre in linear cae or with nonlinear effect included. Algorithm i grounded on the Mawell Equation, implified to the time independent diffuion equation [][4]. If we conider the propagation in one direction and aume the low variation of the inde along the z-ai and low angle divergence of the light beam in linear medium, we obtain the net wave equation: ( n n ) k E E ( n E) E E + 0 in k n 0 z () where: E i the electrical field vector, k i the wave number of beam, n i the refractive inde of the medium, and n 0 i the reference refractive inde. B olving equation () one can calculate amplitude and phae ditribution of the field. However in order to utilized implification thi equation can be ued onl without taking into account backward reflection and for beam which propagate in direction at relativel mall angle to the z ai about 0 degree maimum... Numerical implementation of the method The developed program i written in FORTRAN 95 language [3], which guarantie low numerical cot of computation. The computational area i dicretized. The field i repreented b a regular grid of point eparated on the ditance along t he t-ai, tand t talong t he t-ai. taccording to t hat, tpoint tp i,j ha coordinate (i, j ). Denit of the grid point define the computation accurac and efficienc. The higher denit reult in more accurate, but more time conuming calculation. Thu it i necear to achieve the agreement between computation accurac and efficienc. * phone: (+48 ) , M.Zdanowicz@itl.waw.pl, Lightguide and Their Application III, edited b Jan Wójcik, Waldemar Wójcik, Proc. of SPIE Vol. 6608, 66080S, (007) X/07/$8 doi: 0.7/ Proc. of SPIE Vol S- Downloaded From: on 0/9/06 Term of Ue:

2 Equation () i tranformed in according to the ued finite difference method. In conequence we obtain two equation for both component of the electrical field: E and E. Thee two equation are ued for computation in the program. Generall the can be introduced a the tem of matri equation: E E + + [ M ] ([ N ] E + [ Q ] E ) [ M ] ([ N ] E + [ Q ] E ) where: M, N, Q are the tri-diagonal matrice containing coefficient for each calculated point and E and E repreent two component of the electrical field vector. The electrical field in the net tep (+, z + z +h, h i the tep ize) i calculated with the ue of the filed ditribution known from the previou tep (, z )..3. Nonlinearit The effect of the nonlinearit can be taken into account b the nonlinear term in refractive inde: () n n n E (3) L + NL where: n i the total refractive inde, n L i the linear refractive inde, n NL i the nonlinear coefficient. Here the Kerr nonlinearit i conidered, which i the mot common in optical fibre. Definition (3) decribe the dependence of the total refractive inde on the intenit of the propagating field. It mean that the field elf modifie the propertie of the medium without eternal effect. Nonlinear Kerr effect provide additional impact on the beam propagation. The mot intereting i the effect of elf focuing, which in pecific condition can reult in formation of the patial oliton when the elf focuing compenate the effect of diffraction widening. In order to increae accurac of the performed calculation we ue the plit tep local error method. Thi method allow controlling the tep ize uing a meaure of the local error. The cheme implemented in preented program utilie the technique of the tep doubling and local etrapolation. At firt the field ditribution at z+h i calculated; the obtained olution i o called coare olution, e c. Net, the program return to z and compute the fine olution, e f, with ue of two tep of h ize. The appropriate linear combination of coare and fine olution give approimate field ditribution E at the z+h. Thi olution i given b epreion: E 4 e f e c 3 3 (4) In the local error method the tep ize i choen adaptivel o that the local error incurred from z to z+h i bound within a pecified range. The relative local error i defined a: K, L f i, j L [ e ( i, j) e ( i, j) ] K, [ e f ( i, j) ] i, j c σ (5) where: K i the total number of grid point along the -ai, L i the total number of grid point along the -ai, e f (i,j), e c (i,j) are the fine and coare field intenit value in grid point (i,j), repectivel. Equation (5) give the mean quadratic deviation of the fine and coare olution. The tep ize i choen b keeping the relative local error value σ in the pecified range (/σ G, σ G ), determined b a fied value σ G the goal local error. If σ > σ G the obtained olution i dicarded and tep ize i halved. If σ i in the range (σ G, σ G ) the tep ize i divided b a factor of /3 for the net tep. If σ < /σ G, the tep ize i multiplied b a factor of /3 for the net tep. Field ditribution obtained from the equation (4) i the input for net tep computation. Proc. of SPIE Vol S- Downloaded From: on 0/9/06 Term of Ue:

3 . NUMERICAL RESULTS Uing the propoed algorithm we imulate light beam propagation in olid core microtructure fibre with heagonal lattice of air hole. The input data are contained in two tet file. The firt one contain parameter of the imulated tructure, like the pitch D, air hole diameter φ or the number of the air hole ring urrounding the core. The econd file contain parameter of the input field, propagation ditance and tep ize along the, and z-ai I. - I ' I Fig.. Eample of refractive inde ditribution ued during imulation. Reult of the computation are written into three tet output file: the firt two file contain the component of the field E, E and the third one contain the module of the field E. Thee three file are aved everal time for equal egment of the propagation length. Net, the obtained data file are ued to plot the field ditribution with the ue of GnuPlot program. The eample of the inde ditribution acro the fibre under invetigation i hown in the figure. Baic parameter of that tructure can be eail modified in the input tet file. Fig.. Input ignal Gauian beam. The input field ditribution i a Gauian beam. An eample of field ditribution i hown in figure. Thi ignal propagate along the z-ai of the preented tructure (fig. ). The refraction inde and the input field are encloed in two matrice which are ued for calculation of the field ditribution in the firt tep. The calculated field i ued a the input ignal for computation in the net tep. Proc. of SPIE Vol S-3 Downloaded From: on 0/9/06 Term of Ue:

4 z 0 µm z 0 µm z 40 µm z 40 µm Fig. 3. Evolution of the field inide PCF in linear (left ide) and nonlinear cae (right ide). Normalized intenit,5,4,3,, 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0, 0, , Cro ection of the beam in direction [µm] Nonlinear cae Linear cae Input beam Fig. 4. Comparion of the beam hape at z 40 µm in linear and nonlinear cae. Intenit i normalized to the input value. Proc. of SPIE Vol S-4 Downloaded From: on 0/9/06 Term of Ue:

5 Figure 3 how the evolution of the propagating field in the ame tructure in linear cae (left) and including the nonlinear effect (right). Although both beam look imilar, in nonlinear cae one can oberve lightl tronger and better localized field. Figure 4 how the cro ection of the imulated beam in both cae at the range of z 40µm. Thi figure how better the influence of the elf-focuing effect. In linear cae onl two factor decide the beam propagation mechanim. The characteritic of the fibre tructure and the diffraction effect, cauing beam broadening and decreae of intenit. In cae of propagation with the nonlinear effect, the elf-focuing effect can be oberved. For a pecific group of input parameter, a patial oliton can be formed in the photonic crtal fibre. Thi i poible when the diperion effect are balanced with the nonlinear elf-focuing. The knowledge of the eact nonlinear effect influence of the field propagation in PCF can be ued for reearch and deign of new device for micro-optic. 3. FUTURE WORK AND CONCLUSIONS The preented computer program i ued for imulating beam propagation in the frequenc regime in PCF with heagonal tructure. Uing thi program i intended to eamine the reflection on the connection urface between conventional and photonic crtal fibre, but thi reearch demand ome further modification of the algorithm. The developed program can be a ground for future numerical tool. With imple modification thi algorithm can be applied for the imulation of the tructure with hole filled b a material with a pecified characteritic. It i planned to develop an algorithm to create the inde value matrice from the real photonic tructure, a an opportunit to invetigate a real tructure, with added poibilit of changing parameter of material ued. REFERENCES. O.V. Sinkin, R. Holzlöhner, J. Zweck, C.R. Menuk: Optimization of the Split-Step Fourier Method in Modeling Optical Fiber Communication Stem, Journal of Lightwave Technolog, vol., No., Januar L. Adamowicz, Van Q. Nguen: Electromagnetic field in a lab of photonic crtal b BPM, Optic and Laer in Engineering, No. 35, pp , W.H. Pre, S.A. Teukolk, W.T. Vetterling, B.P. Flanner: Numerical Recipe in Fortran 77. The Art of Scientific Computing. Second Edition, Cambridge Univerit Pre, F. Fogli, L. Saccomandi, P. Bai, G. Bellanca, S. Trillo: Full Vectorial BPM Modeling of Inde Guiding Photonic Crtal Fiber and Coupler, Optic Epre, vol. 0, No., 00. Proc. of SPIE Vol S-5 Downloaded From: on 0/9/06 Term of Ue: