Stable Soliton Propagation in a System with Spectral Filtering and Nonlinear Gain

Transcription

1  Fiber and Integrated Optic, 19:31] 41, 000 Copyright Q 000 Taylor & Franci r00 $1.00 q.00 Stable Soliton Propagation in a Sytem with Spectral Filtering and Nonlinear Gain  MARIO F. S. FERREIRA Ä MARGARIDA M. V. FACAO SOFIA C. V. LATAS Department of Phyic Univerity of Aveiro Aveiro, Portugal Stable oliton propagation in a ytem with linear and nonlinear gain and pectral filtering i invetigated. Different type of exact analytical olution of the cubic and the quintic complex G inzburg-landau equation ( CGLE) are reviewed. The condition to achieve table oliton propagation are analyzed within the domain of validity of oliton perturbation theory. We derive an analytical expreion defining the region in the parameter pace where table pulelike olution exit, which agree with the numerical reult obtained by other author. An analytical expreion for the oliton amplitude correponding to the quintic CGLE i alo obtained. We how that the minimum value of thi amplitude depend only on the ratio between the linear gain and the quintic gain aturating term. Keyword nonlinear optic, optical fiber communication, optical oliton The ue of narrow-band filter in optical oliton tranmiion ytem ha beneficial effect. For example, the diffuion of oliton center frequency caued by a uperpoition of amplifier noie ( the Gordon-Hau effect w1x) i uppree d w, 3 x, the oliton amplitude i tabilized w3, 4 x, and interaction between oliton i reduced w5] 7 x. The Raman elf-fre quency hift can alo be uppre ed by the action of narrow-band filter w8, 9 x. When the filter are ued, ome exce gain mut be provided around the filter center frequency to compenate for the lo that oliton uffe r at the wing of their pectra. The exce gain amplifie line ar wave coexitent with oliton train, leading to intability of the background. The untable linear wave degrade the ignal-to-noie ratio, and if their power grow comparable to that of the oliton, the oliton may be detroyed w10, 11 x. The intability caued by the accumulation and amplification of the background linear wave can be uppre ed by liding the center frequency of filter, whereby the tranmiion line i made opaque to the linear wave w1, 13 x. Another method conit of uing nonline ar gain ( the amplitude-dependent gain ), which preferentially amplifie the oliton with large amplitude while the line ar wave with mall amplitude are unamplified or attenuated w14 x. Received 3 April 1999; accepted 4 May Addre correpondence to Mario Fernando do Santo Ferreira, Department of Phyic, Univerity of Aveiro, 3800 Aveiro, Portugal. MFERREIRA@ fi.ua.pt 31

2 È d 3 M. F. S. Ferreira et al. In thi paper we tudy oliton propagation in the preence of pectral filtering and linear and nonline ar gain. In the next ection we preent ome exact analytical reult concerning pule olution of the cubic and quintic CGLE. Then we derive the evolution equation for the oliton parameter uing oliton perturbation theory and dicu the exitence and characteritic of table oliton olution of the quintic CGLE with fixed amplitude. The final ection ummarize the main concluion. Exact Analytical Reult The pule propagation in optical fiber where line ar and nonline ar amplifier and narrow-band filter are periodically inerted may be decribed by the following modified nonline ar Schrodinge r equation if the inertion period of thee device i ufficiently maller than the diperion ditance w14, 15 x: q 1 q q 4 i q q < q < q id q q ib q i«< q < q q im < q < q ( 1) Z T T where Z i the propagation ditance, T i the retarded time, q i the normalized envelope of the electric field, b tand for pectral filtering ( b ) 0 ), d i the linear gain or lo coefficient, «account for nonline ar gain-aborption procee, and m repreent a higher order correction to the nonline ar gain aborption. Equation ( 1) i alo known a the complex Ginzburg-Landau equation ( CGLE ), o-called cubic for m 0 and quintic for m / 0. We will conider firt the cubic cae and aume a tationary olution of Eq. ( 1) in the form v w x 4 q T,Z a T exp id ln a T y iv Z where a ( T ) i a real function and d, v are real contant. By inerting Eq. into Eq. ( 1) ( with m 0 ), the following olution can be obtained for a ( T ) w16, 17 x: where a T A ech BT 3 and d i given in the form ( B y d ) A q 3db B ( 4) B ( 5) b d q d y b 3( 1 q «b ) y 9( 1 q «b ) q 8 ( «y b ) d ( 6) ( «y b ) On the other hand, we have ( d 1 y d q 4b d) v y ( 7) ( d y b q b d )

3 d b Stable Soliton Propagation 33 The olution given by Eq. 3 ha a ingularity at d y b q b d 0, which take place on the following line in the plane ( b,«): b 3 1 q 4b y 1 ««( 8) q 9b The line given by Eq. 8 i repreented in Figure 1 olid curve and ha the following limiting value: «b < 1 ( 9a ) 1 «b 4 1 ( 9b ) 3 It can be hown that for d ) 0 the olution given by Eq. ( 3) exit and i table below the curve given by Eq. ( 8 ). However, the background tate i untable. For - 0 the olution given by Eq. ( 3) exit above the curve Eq. ( 8 ), but it i untable w17 x. If b and «atify Eq. ( 8) and d 0, a olution of the cubic CGLE with arbitrary amplitude exit w18, 19 x, given by a T C ech DT 10 where C i an arbitrary poitive parameter and C r D i given by We have alo ( ) q 9b 1 q 4b 1 q 4b y b ( 3 1 q 4b y 1) C D 1 q 4b y 1 1 q 4b d v yd D 1 b b Figure 1. Curve «( left cale ), given by Eq. ( 8 ), and amplitude-width product C r D for the arbitrary-amplitude oliton ( right cale) veru filtering parameter b.

4 34 M. F. S. Ferreira et al. Figure 1 how the amplitude-width product C r D veru b calculated along the pecial line given by Eq. ( 8 ). The limiting value of the amplitude-width product A r B for the fixed-amplitude oliton coincide with the value C r D on the line given by Eq. ( 8 ). Thi how that arbitrary-amplitude oliton can be conide red a a limiting cae of fixed-amplitude oliton when d ª 0. However, the arbitrary-amplitude oliton have tability propertie differe nt from thoe for fixed-amplitude oliton. In fact, it wa hown that arbitrary-amplitude oliton are table pule, which propagate in a table background becaue d 0 w15 x. Fixed-amplitude oliton can alo be found in the cae of the quintic CGLE, but they appear to be untable at every point of the parameter pace w15 x. On the other hand, if b and «atify Eq. ( 8) and we have d 0, a table olution with arbitrary amplitude alo exit for the quintic CGLE w18, 19 x, given by 3d( 1 q 4b ) P f ( T ) w a ( T )x ( 13) ( b y «) q S coh( P T ) where P i an arbitrary poitive parameter and 9d m ( 1 q 4b ) 3b y d y b d S ( b y «) q P ( 14) 1 q 4b y 1 1 q 4b d v yd P 15 b b When m ª 0, the olution Eq. ( 13) tranform to the arbitrary-amplitude olution of the cubic CGLE, given by Eq. ( 10 )] ( 1 ). Reult of Soliton Perturbation Theory Auming that all the coefficient on the right-hand ide of Eq. ( 1) are mall, we can ue the adiabatic oliton perturbation theory w0, 1x to evaluate the dynamical evolution of the oliton parameter, the amplitude h and the frequency k, with which the one-oliton olution i given by v w x 4 q T,Z h Z ech h Z T q k Z y u i w 5 = exp yik ( Z ) T q h ( Z ) y k ( Z ) Z y i ( 16) Applying the perturbation procedure, we get the following et of ordinary differential equation: dh d h y b h h q k q «h q m h ( 17) dz dk 4 y b h k ( 18) dz 3 A can be een from Eq. ( 18 ), the oliton frequency approache aymptotically to k 0 ( table fixed point) if h ` 0. The table fixed point for the oliton

5 Stable Soliton Propagation 35 amplitude, on the other hand, are given by minimum of the potential function f, defined by dh df y ( 19) dz dh Conidering Eq. 17, we have the following expreion for the potential function: f h yd h q b y «h y m h ( 0) 6 45 For the zero-amplitude tate to be table, the potential function given by Eq. ( 0) mut have a minimum or, at leat, to be locally contant at h 0, in addition to a minimum at h h ` 0. Thee objective can be achieved if the following condition are verified: 4 d F 0 m - 0 «) b r 15d ) 8m h 1 We can verify from the above condition that the incluion of the quintic term in Eq. ( 1) i neceary to have the double minimum potential. The tationary value for the oliton amplitude can be obtained from Eq. ( 0) and i given by y5 ( «y «) y 5 ( «y «) y 4d m r5 h 8m where «i given by Eq. ( 9a ). However, the reult given by Eq. can be generalized for arbitrary value of b uing «given by Eq. 8 w x. Thi more general reult can be verified introducing Eq. and ( 10) into Eq. ( 1 ), which how that the table pulelike olution of the quintic CGLE belong to the ame family of olution a the arbitrary-amplitude olution of the cubic CGLE. The dicriminant in Eq. mut be greater than or equal to zero for the olution to exit. For given value of b, m, and «, the allowed value of d to guarantee a table pule propagation mut atify the condition d min F d F 0, where 5 ( «y «) d ( 3) min 4m When d 0, the peak amplitude i found to achieve a maximum value: 5 ( «y «) h y ( 4) max 4 m For m 0 and ««thi peak amplitude become arbitrary, a oberved in the previou ection for the cae of the cubic CGLE. On the other hand, for given value of b, d, and m, the minimum value of allowed «become ««q 4d m r5 ( 5) min

6 36 M. F. S. Ferreira et al. We can verify from the lat condition in Eq. ( 1) or, alternatively, from Eq. and ( 5 ), that there i a minimum value for the peak amplitude, given by 4 15d h ( 6) min 8m Thi minimum value i determined uniquely by the quotient between the line ar exce gain rlo and the quintic aturating gain term. From Eq. ( ), we obtain a tationary amplitude h 1 when 15d q 8m ««y ( 7) 1 10 Figure how the potential given by Eq. ( 0) for d y 0.05 ( olid curve) and m y0.5 ( curve a ), m y ( curve b ), and m y0.5 ( curve c ). The dahed curve correpond to d y0.1 and m y0.5 ( curve a9 ) and m y0.5 ( curve c9 ). In all cae we conider b 0.3 and «0.5. It can be een from Figure that the tationary amplitude h incre ae when < m < decreae. Curve b corre- pond to the cae h 1, which occur when the coefficient on the right-hand ide of Eq. ( 1) atify the condition Eq. ( 7) with «b r. In the cae of curve a9 there i no minimum of the potential function for h ` 0, ince the condition Eq. ( 5) i not atified. Figure 3 how the potential function given by Eq. ( 0) when the relation Eq. ( 7) i atified for b 0.3, «0.5, m y0.5 ( curve a ), m y ( curve b ), and m y0.5 ( curve c ). Curve b and c preent a minimum at h 1 and h 0, ince they atify the condition Eq. ( 3) and correpond to negative value of the linear gain ( d y 0.05 and y 0.1, repectively ). However, curve a ha no minimum at h 0, ince the correponding linear gain i poitive ( d ). Figure 4 illutrate the tability characteritic of the table olution uing the phae-plane formalim. Figure 4 A correpond to curve a in Figure 3, and we f h b «d m m m d m m Figure. Potential veru oliton amplitude for 0.3 and 0.5. Solid curve correpond to y0.05 and y0.5 ( curve a ), y ( curve b ), and y 0.5 ( curve c ). Dahed curve correpond to y0.1 and y 0.5 ( curve a9 ) and y 0.5 ( curve c9 ).

7 Stable Soliton Propagation 37 Figure 3. Potential f veru oliton amplitude h when the relation Eq. ( 7) i atified for b 0.3, «0.5, m y 0.5 ( curve a ), m y ( curve b ), and m y 0.5 ( curve c ). Figure 4. Phae portrait of Eq. 17 and 18 correponding to A curve a and B curve b of Figure 3.

8 38 M. F. S. Ferreira et al. oberve that, in thi cae, oliton propagation can be affected by background intability due to the amplification of mall-amplitude wave. An intereting feature of Figure 4 A i the limited bain of attraction of the teady tate olution. For example, initial condition with h i 0.7 and k i " 1 evolve toward the trivial olution h 0 of Eq. ( 17) and ( 18 ). For thee initial condition, the nonline arity i not ufficiently trong to balance diperion, and the pule dipere away. The dahed curve in Figure 4 A give approximate limit between differe nt bain of attraction. From a perturbation analyi of Eq. ( 17) and ( 18) around h 0, one can how that thee curve cro the h 0 axi at k c " Thu wave with weak initial amplitude grow up to h 1 if < k i< In thi cae, oliton propagation can be everely affected by the background intability. Figure 4 B correpond to curve b in Figure 3, and we can ee that, in thi cae, the background intability i avoided, ince the mall-amplitude wave are attenuated, irrepective of their frequency k. Figure 5 how, the dependence of «( dahed curve ), «( dotted curve ) min 1, and for reference, «( olid curve) on b for d y0.0, m y0.1, and m y0.5. The reult given by Eq. ( 5) explain the numerical olution hown in Ref. w15 x, namely, that in ( b,«) the lower limit of the region at which table pulelike olution of the quintic CGLE are found i almot parallel to the line «and that a < m < or < d < incre ae, thi lower limit alo increae. The dependence of the peak intenity h on the quintic aturating gain term m i illutrated in Figure 6 for b 0.4, d y0.01, «0.3 ( curve a ), and «0.5 curve b. We oberve that h increae and tend to infinity when < m < ª 0. It alo increae when «incre ae and ror b decre ae. Concluion In thi paper we have invetigated the condition to achieve table oliton propagation in a ytem with linear and nonlinear gain and pectral filtering. We conidered different type of olution of the cubic and the quintic complex Ginzburg- Landau equation, namely, olution with fixed amplitude and olution with arbi- Figure 5. Dependence of «( dahed curve ), «( dotted curve ), and «( olid curve) min 1 on filtering parameter b for d y 0.0, m y 0.1, and m y 0.5.

9 Stable Soliton Propagation 39 Figure 6. Peak intenity h veru quintic aturating gain term m for b 0.4, d y 0.01, «0.3 ( curve a ), and «0.5 ( curve b ). trary amplitude. Thee arbitrary-amplitude olution correpond to table oliton, which exit on pecial line in the parameter pace where olution with fixed amplitude become ingular. In the cae of the cubic CGLE they form the only table cla among all the tationary pule. In the cae of the quintic CGLE the cla of arbitrary-amplitude oliton i alo table. Moreover, we have alo found the condition for the table propagation of fixed-amplitude oliton of the quintic CGLE within the domain of validity of perturbation theory. Thee olution belong to the ame family of olution a the arbitrary-amplitude olution of the cubic CGLE. We derived alo an expreion for the lower limit of the region in the plane ( b,«) at which table pulelike olution of the quintic CLGE can be obtained, corroborating the numerical reult reporte d previouly by other author. In addition, a minimum value for the pe ak amplitude of the table olution wa found, which depend unique ly on the quotie nt between the linear exce gain and the quintic aturating gain term. Our reult can be ueful in determining the value of ytem parameter required to obtain oliton with given characteritic. Reference 1. Gordon, J. P., and H. A. Hau Random walk of coherently amplified oliton in optical fibre tranmiion. Opt. Lett. 11:665.. Kodama, Y., and A. Haegawa Generation of aymptotically table optical oliton and uppreion of the Gordon-Hau effect. Opt. Lett. 17: Mecozzi, A., J. D. Moore, H. A. Hau, and Y. Lai Soliton tranmiion control. Opt. Lett. 16: Kubota, H., and M. Nakazawa Soliton tranmiion control in time and frequency domain. IEEE J. Quant. Electron. 9: Afanajev, V. V Interpretation of the effect of reduction of oliton interaction by bandwidth-limited amplification. Opt. Lett. 18: Ferreira, M. F Analyi of oliton tability and interaction in a ytem with nonlinear gain. SPIE Proc. 994:349.

10 Â È 40 M. F. S. Ferreira et al. 7. Okamawari, T., and A. Haegawa Analyi of oliton interaction by mean of a perturbed invere-cattering tranform. Phy. Rev. A. 51: Nakazawa, M., K. Kurokawa, H. Kubota, and E. Yamada Obervation of the trapping of an optical oliton by adiabatic gain narrowing and it ecape. Phy. Rev. Lett. 65: Ferreira, M. F Analyi of femtoecond optical oliton amplification in fiber amplifier. Opt. Commun. 107: Kodama, Y., M. Romagnoli, and S. Wabnitz Soliton tability and interaction in fibre laer. Electron. Lett. 8: Matumoto, M., and A. Haegawa Numerical tudy of the reduction of intability in bandwidth-limited amplified oliton tranmiion. Opt. Lett. 18: Mollenauer, L. F., J. P. Gordon, and S. G. Evangelide The liding-frequency guiding filter: An improved form of oliton jitter control. Opt. Lett. 17: Mollenauer, L. F., P. V. Mamyhev, and M. J. Neubelt Meaurement of timing jitter in filter-guided oliton tranmiion at 10 Gbit r and achievement of 375 Gbit r- Mm, error free, at 1.5 and 15 Gbit r. Opt. Lett. 19: Matumoto, M., H. Ikeda, T. Uda, and A. Haegawa Stable oliton tranmiion in the ytem with nonlinear gain. J. Lightwave Technol. 13: Soto-Crepo, J. M., N. N. Akhmediev, and V. V. Afanajev Stability of the pulelike olution of the quintic complex Ginzburg-Landau equation. J. Opt. Soc. Am. B. 13: Pereira, N. R., and L. Stenflo Nonlinear Schrodinger equation including growth and damping. Phy. Fluid 0: Ferreira, M. F Stability condition for oliton fiber laer with nonlinear gain. SPIE Proc. 918: Akhmediev, N. N., and V. V. Afanajev Novel arbitrary amplitude oliton olution of the cubic-quintic complex Ginzburg-Landau equation. Phy. Rev. Lett. 75: Akhmediev, N. N Soliton ingularity in the ytem with nonlinear gain. Opt. Lett. 0: Karpman, K. I., and V. V. Solov ev A perturbational approach to the two-oliton ytem. Phy. D 3: Haegawa, A., and Y. Kodama Soliton in optical communication. Oxford: Oxford Univerity Pre.. Soto-Crepo, J. M., and L. Pequera Analytical approximation of the oliton olution of the quintic complex Ginzburg-Landau equation. Phy. Rev. E 56:788. Biographie Mario F. S. Ferreira wa born in Ovar, Portugal. He graduated in phyic from the Univerity of Porto, Portugal, in He received a Ph.D. in phyic in 199 from the Univerity of Aveiro, Portugal where he i now a profeor in the phyic department. Between 1990 and 1991 he pent 10 month at the Univerity of Eex, Colcheter, England, performing experimental work on external cavity emiconductor laer and nonlinear optical fiber amplifier. At preent, he lead a reearch group dedicated to the modeling and characterization of multiection emiconductor laer for coherent ytem, quantum well laer, optical fiber amplifier and laer, and nonlinear effect and oliton propagation in optical fiber. He ha written more than 10 article for cientific journal and conference publication. Dr. Ferreira i a member of the Portuguee Phyical Society, European Phyical Society, European Optical Society, Optical Society of America, SPIE} The International Society for Optical Engineering, New York Academy of Science, and American Aociation for the Advancement of Science.

11 Ä Â Stable Soliton Propagation 41 Â Sofia C. V. Lata wa born in Evora, Portugal. She received her graduate degree in phyic from the Faculty of Science, Univerity of Libon, in Since then he ha continued tudie at the Phyic Department of the Univerity of Aveiro, where he i an aitant lecturer. Between 1997 and 1998 he wa at the Univerity of Eex, Colcheter, England, and at the Department of Mathematical Modeling of the Technical Univerity of Denmark. At preent, he i performing work toward a Ph.D. Her reearch i primarily concerned with nonlinear effect and oliton propagation in optical fiber. M. Lata i a member of the Portuguee Phyical Society. Margarida M. V. Facao wa born in Ilhavo, Portugal. She received her graduate degree in phyic from the Univerity of Aveiro in Since then, he ha continued tudie at the Phyic Department of the Univerity of Aveiro, where he i an aitant lecturer. In 1993 he pent 5 month at the Dublin College of Technology, performing both experimental and theoretical work in the optic field. At preent, he i working toward her Ph.D. Her reearch i primarily concerned with nonlinear effect and oliton propagation in optical fiber. M. Facao Ä i a member of the Portuguee Phyical Society.