It is shown experimentally and theoretically that a sinusoidally modulated pulse evolves with time into a train of dark soliton-like pulses and then returns to its initial sinusoidal shape on propagation through a nonlinear single-mode fiber with positive group velocity dispersion. The experimental results are in agreement with predictions from the nonlinear Schrodinger equation

Bosshard, C.

Mamyshev, P. V.

Stegeman, G. I.

Wigley, P. G.

Wilson, J.

English

University of Central Florida (UCF): STARS (Showcase of Text, Archives, Research & Scholarship)

University of Central Florida STARS Faculty Bibliography 1990s Faculty Bibliography 1-1-1994 Restoration Of Dual-Frequency Signals With Nonlinear Propagation In Fibers With Positive Group-Velocity Dispersion P. V. Mamyshev University of Central Florida P. G. Wigley University of Central Florida J. Wilson University of Central Florida C. Bosshard University of Central Florida G. I. Stegeman University of Central Florida Find similar works at: https://stars.library.ucf.edu/facultybib1990 University of Central Florida Libraries http://library.ucf.edu This Article is brought to you for free and open access by the Faculty Bibliography at STARS. It has been accepted for inclusion in Faculty Bibliography 1990s by an authorized administrator of STARS. For more information, please contact STARS@ucf.edu. Recommended Citation Mamyshev, P. V.; Wigley, P. G.; Wilson, J.; Bosshard, C.; and Stegeman, G. I., "Restoration Of Dual-Frequency Signals With Nonlinear Propagation In Fibers With Positive Group-Velocity Dispersion" (1994). Faculty Bibliography 1990s. 2961. https://stars.library.ucf.edu/facultybib1990/2961 Appl. Phys. Lett. 64, 3374 (1994); https://doi.org/10.1063/1.111279 64, 3374© 1994 American Institute of Physics.Restoration of dual-frequency signalswith nonlinear propagation in fibers withpositive group velocity dispersionCite as: Appl. Phys. Lett. 64, 3374 (1994); https://doi.org/10.1063/1.111279Submitted: 29 March 1993 . Accepted: 21 April 1994 . Published Online: 04 June 1998P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. StegemanRestoration of dual-frequency signals with nonlinear in fibers with positive group velocity dispersion P. V. Mamyshev, P. G. Wigley, J. Wilson, C. Bosshard, and G. I. Segeman Center for Research in Electra-Optics and Lasers, University of Central Florida, 12424 Research Parkway, Orlando, Florida 32826 (Received 29 March 1993; accepted for publication 21 April 1994) It is shown experimentally and theoretically that a sinusoidally modulated pulse evolves with time into a train of dark soliton-like pulses and then returns to its initial sinusoidal shape on propagation through a nonlinear single-mode fiber with positive group velocity dispersion. The experimental results are in agreement with predictions from the nonlinear Schrodinger equation. Optical signals propagating at power levels for which the nonlinear Schrodinger equation (NSE) is operative exhibit interesting behavior both from a fundamental point of view as well as for practical applications.r-’ The NSE is of special interest because of the existence of soliton solutions, both dark and bright. NSE solitons have been experimentally demonstrated both in the temporall”” and spatial12-15 do- mains. Propagation of laser radiation through single-mode optical fibers appears to be the best physical example of a system described by the NSE.3 Modulational instability in fibers due to a dual-frequency input signal (with unequal relative intensities of the spectral components) has been ex- perimentally investigated in the negative group velocity dis- persion spectral region. It led to the formation of trains of bright soliton-like pulses in the first stage of the signal propagation.4 The generation of a stable train of fundamental solitons from a dual-frequency input signal in nonuniform fibers with slowly decreasing value of the dispersion has also been demonstrated experimentally? One of the interesting theoretical predictions of the NSE is the restoration with propagation of some periodic signals and higher-order soliton pulses.‘~‘-* The restoration of higher-order soliton pulses over a soliton period has been experimentally observed,r6 but to the best of our knowledge this effect has not yet been studied experimentally for the case of periodic input signals. Here we investigate the resto- ration phenomena of a dual-frequency signal E(z=O,t)=Eo cos y exp[-i Ff] (1) in the region of positive group velocity dispersion, where Ao=or-~a is the frequency difference between the input frequencies. We show both theoretically and experimentally that this periodic input signal evolves into a train of dark soliton-like pulses, and then with further propagation returns to a sinusoidally modulated pulse. The nonlinear Schrijdinger equation is given by’ i $A?~$A+A~A[‘=O, which has nonlinear eigemnodes known as bright (+ sign) and dark (- sign) solitons. In this case the dimensionless parameters are as follows: E(z,t) =L(t,z)exp(ikz- iwof) is the electric field; o. is the center frequency (w,+u&2; z is the axial fiber coordinate; t is time;-k=aowolc is the propa- gation constant; A(&v)=A(z,t)/A,; ~~=2ck,l(t&10~n~); n=no+n,[E]’ is the refractive index; t=z/ztisp; zdt,=t$lli,l is the dispersion length; ~~=(t-&z)lto; f. is a normalization parameter; ki=a’k/dti’; and the sign in Eq. (2) is opposite to the sign of the fiber group velocity (or second-order) dispersion k2. Note that the signal given by Eq. (2) is written in dimensionless units and the normaliza- tion parameter to in Eq. (2) is chosen to be a duration [full width half-maximum (FWHM) intensity] of the input signal beat cycle. In dimensionless units the input signal is A(O,v)=A, cos i i m; . (3) Theory6p7 predicts periodic (or quasiperiodic) with normal- ized distance 6 behavior for such a signal, shown in Fig. 1 by the dashed line in (a), (c), and (e). This periodic behavior was investigated with a mode- locked NaCl color center laser operating in the wavelength range h=1.53-1.54 pm and a 0.9 km long, low-loss (0.25 dB/km), single-mode, dispersion-shifted fiber (zero disper- sion point x=1.555 ,um). Because the fiber was dispersion shifted, the group velocity dispersion, which would be nega- tive in a normal fiber, exhibits positive group velocity dis- persion (k,>O) for the wavelengths used. By appropriately adjusting the cavity, the laser generated dual-frequency pulses of 10-20 ps duration. The intensities of the spectral components were equal. The separation between the two spectral components Aw/(27rc)=1/(2toc j could be varied from 5-22 cm-‘, which corresponded to a temporal sinu- soidal pulse modulation of 150-600 GHz. The intensity au- tocorrelation function of the pulses showed a sinusoidal modulation with a contrast of 3:l [Figs. l(a) and l(b); Fig. 2(a)]. The fact that we used dual-frequency pulses instead of a continuous-wave dual-frequency signal, means that A, in Eq. (3) is not a constant but a pulse duration is much larger than 1 in our normalized units. In this spectral region and with our experimental input parameters, Raman self- scattering effectsI are of no importance for propagation of our signal, and Eq. (2) is still valid. [In the negative group- velocity-dispersion spectral region, however, the Raman self- scattering effect can dramatically change the evolution of the dual-frequency signal from that predicted by Eq. (2)l*.] The laser radiation was coupled into the fiber with 10X objective, and the input power was varied by combination of 3374 Appl. Phys. Lett. 64 (25), 20 June 1994 0003-6951/94/64(25)/3374/3/$6.00 0 1994 American Institute of Physics time, 77 FIG. 1. Evolution of a dual frequency signal in a fiber. Left column- numerical simulations, right column-experiment. Dashed lines-intensity envelop- ~Gd12/lA~12, solid lines-intensity autocorrelation functions. The dimensionless parameters are A,=4, first row (a) and (b) PO; second row (c) and (d) E=O.14; third row (e) and (f) 6=0.4. Experimental parameters: fiber length 0.9 km; (c) and (d) Ao/(27rc)=S.S cm-r, input intensity=2X106 W/cm’; (e) and Q Ao/(27rc)=ll cm-‘, input intensity=6XlO” W/cm”. FIG. 2. Evolution of the frequency spectrum in a fiber. Solid lines- experiment, dashed lines--numerical simulations. (a), (b), and (c) are the spectra of the signal shown, corresponding to the tirst, second, and third row of Fig. 1. the rotation of a half-wave plate followed by a polarizer. By changing the frequency separation between the two laser spectral components Aw (i.e., by changing the modulation frequency of the input signal) we could change the dimen- sionless length of the 0.9~km-long fiber. (The dimensionless fiber length could also be changed by tuning the laser wave- length via the frequency dependence of the fiber dispersion.) This means that by changing the modulation period of the signal and correspondingly adjusting the input intensity (so that the dimensionless input intensity remained constant), we could effectively investigate the spatial evolution of the dual- frequency signal without changing the actual physical fiber length. The temporal characteristics of the output signal were determined by measuring the background-free intensity au- tocorrelation functions. Output spectra were measured with a scanning Fabry-Perot interferometer. At low input intensities the effects of the nonlinearity are negligible for the fiber length used and the signal propagates (almost) without change of its spectral and temporal enve- lopes. In our normalized units, a low intensity input corre- sponds to a dimensionless peak intensity for which the input sinusoidal modulation period is much smaller than that re- quired for a fundamental soliton, that is, Aoel. Nonlinear effects become important at higher input intensities, i.e., when A0 is of the order of, or larger than 1. Experimental results for the case of A0 =4 are shown in Figs. 1 and 2. For comparison, results of numerical simulations are also shown in these figures. New spectral sidebands separated by Aw are generated due to parametric processes at the early stages of the propagation through the fiber [Fig. 2(b)].1231g Their rela- tive phases are such that the signal is reshaped into a train of dark pulses in the time domain [Figs. l(c) and l(d)]. There are rr phase jumps across the minima of the dark pulses, a feature characteristic of dark solitons. Nevertheless, the pulses are not stable with respect to further propagation in the fiber as shown in Figs. l(e) and l(f). To understand quali- tatively the reason for the instability of these soliton-like pulses, one can find from numerical simulations that (i) the pulse background is chirped, and (ii) the background inten- sity lAbI is less than that needed for dark solitons with the same pulse width C- (FWBM). That is, the area of the dark pulses, defined as ~=0.322/A~j’~~, is less than 1. Note that for clean dark solitons the background should be unchirped and the dark pulse area should be s = 1. With further propa- gation the spectral sidebands disappear [Fig. 2(c)] and the signal almost returns to its initial sinusoidal shape [Figs. l(e) and l(f)]. The two spectral components of the restored signal are broadened by self-phase modulation at the fiber output [Fig. 2(c)], because we used a pulsed input signal instead of a cw one. Note that our numerical simulations show that for a cw input signal the recurrence phenomenon occurs at C-1, and at the fiber length corresponding to t-O.4 an approxi- mate recurrence takes place. In conclusion, we have investigated experimentally the propagation of a dual-frequency signal through a single mode fiber in the positive group velocity dispersion spectral region. The results confirm theoretical predictions that the dual-frequency signal propagating in a physical system de- scribed by the “self-defocusing” nonlinear Schriidinger equation first evolves into a train of dark soliton-like pulses, and then evolves back into almost its initial sinusoidal shape. The two spectral components of the restored signal are broadened in comparison with the input spectral compo- nents. This broadening is due to self-phase modulation in the Appl. Phys. Lett., Vol. 64, No. 25, 20 June IQ94 Mamyshev et a/. 3375 fiber, because we used a pulsed dual-frequency signal instead of a cw one. This research was supported by AFOSR (91-0339). We thank Dr. N. Akhmediev and Dr. S. Trill0 for helpful discus- sions. ‘V. E. Zakharov and A. B. Shabat, Sov. Phys. JJXP 34, 62 (1972); V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37,823 (1973). ‘Solitons in Action, edited by K. Lonngren and A. Scott (Academic, New York, 1978). 3A. Hasegawa and F. Tappert, Appl. Phys. L&t. 23, 142, 171 (1973). ‘K. Thi, A. Tomita, J. L. Jewell, and A. Hasegawa, Aoul. Phvs. Lett. 49, L A  - 236 (1986). 5N. N. Akhmediev, V. M. Eleonskii, N. E. Kulagin, Zh. E!ksp. Teor. Fiz. 89, 1542 f1985’) TSov. Phvs. JETP 62, 894 (1985)]; Tear. Mat. Fiz. 72, 809 (1987)‘ [Theor. Math. irnys. 72, 183 (1987)]. 6J. R. Thompson and R. Roy, Phys. Rev. A 43,4987 (1991). 7C. S. West and T. A. Kennedy, Phys. Rev. A 47, 12.52 (1993). *N. N. Akhmediev and A. Ankiewicz, Phys. Rev. A 47,3213 (1993). ‘P. V. Mamyshev, S. V. Chemikov, E. M. Dianov, IEEE J. Quantum Eleo tron. 27, 2347 (1991). “L. E Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. L&t. 45,1095 (1980). %, M. Weiner, J. P. Heritage, R J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Ton&son, Phys. Rev. Lett. 61, 2445 (1988). I’M. Shalaby and A. J. Barthelemy, IEEE I. Quantum Electron. 28, 2736 (1992). 13G. A. Swartzlander, Jr., D. R. Anderson, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991). 14S. k. Skinner, G. R. AIlan, D. R. Anderson, and A. L. Smirl, IEEE J. Ouantum. Electron. 27, 2211- (1991). =JI S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jaeckel, E. M. Vogel, and P. W. E. Smith, J. Opt. SOC. Am. B 8, 1290 16!.9g)&olen, L. F. Mollenauer, and W. J. Tomlinson, Opt. Lett. 8, 186 (1983). “P. V Mamyshev, in Optical Solitons-Theory and Experiment, edited by J. R. Taylor (Cambridge University Press, Cambridge, 1992), Chap. 8, pp. 226~313. 18P V Mamyshev, S. V. Chernikov, E. M. Dianov, and A. M. Prokhorov, &t:Lett. 15, 1365 (199Oj. 19k 0. Hill, D. C. Johnson, B. S. 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Restoration Of Dual-Frequency Signals With Nonlinear Propagation In Fibers With Positive Group-Velocity Dispersion

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Restoration Of Dual-Frequency Signals With Nonlinear Propagation In Fibers With Positive Group-Velocity Dispersion

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