B. Initial Value Problems of One-Dimensional Self-Modulation of Nonlinear Waves in Dispersive Media

Abstract

We propose a novel Er-doped fiber laser with an adjustable output pulse. The optical system is composed of an integrated optic circuit of LiNbO, waveguide, a Sagnac fiber loop, and an Er-doped fiber. A simple analysis is presented and a related experiment is conducted. Our result shows that when the mode lock occuls both the pulse width and the pulse repetition can be simply adjusted by the amplitude of the phase modulation signal. In addition, this fiber laser can generate an output with unequal separations of optical pulse, and these separations are also adjustable. 0 1996 John Wiley & Sons, Inc.Recently, there has been a considerable interest in developing rare-earth-doped fiber lasers for applications in opticalfiber communications and optical sensors. The laser operation can be chosen to be single line [l], multiline [2], or pulsed [3]. For generation of the optical pulse with Er-doped fiber, there are two common methods. The first is the active mode-locked method with an intensity/phase modulator within the fiber ring [4]. Another method is passive mode lock with the use of the nonlinear Kerr effect in the fiber, which 26MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol 12, includes the nonlinear polarization rotation [5], or a nonlinear fiber-loop mirror [6]. In this Letter, we propose a novel Er-doped fiber laser to generate an output pulse that can be simply adjusted by controlling the amplitude voltage of an applied phase modulation signal. Figure 1 illustrates the basic construction and experimental setup. At first, the optical wave generated from the ASE of an Er-doped fiber, which is pumped by two 1.48-pm LDs, is injected into the sensing loop through a 3-dB fiber coupler and an integrated optic circuit. The light wave is then separated into two parts by the Y-branch LiNbO, integrated optic circuit. In the meantime the opposite-sign phase modulation is produced for these two light waves. After propagation through the polarization-maintaining fiber of the Sagnac loop with clockwise (CW) and counterclockwise (CCW) waves, the two phase modulated optical waves interfere and the interfering signal returns from the integrated optic circuit. Please note that this returned interfering signal has a wavelike form of intensity modulation because of the time difference and the opposite sign of phase modulation between CW and CCW waves. This interfering signal is then amplified by the fiber amplifier and fed back to the Sagnac loop through the fiber coupler and integrated optic circuit. If the gain of the fiber amplifier is sufficient to compensate for the losses in the whole system (including the port coupling loss of the fiber coupler and the Y branch), laser emission can be produced. Furthermore, if the frequency of the phase modulation signal and the length of whole fiber loop L (including both the sensing loop and feedback loop of the Er-doped fiber amplifier) satisfies the relation of W T = 2 i n (i = 1,2,. . . , T is the transmission time of the total round-trip, including the time on both the Sagnac ...

Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation

“…The class of Tovbis-Venakides potentials (see (8)) involves a parameter µ ∈ R (µ = 0 is the special case studied earlier by Satsuma and Yajima [19]). In fact, we chose to use the symbol µ for the parameter in (2) precisely because this parameter plays a similar role.…”

“…Satsuma and Yajima [19] found that with initial data q(x, 0; N )=N sech(x) the scattering data relevant for the focusing NLS equation can be found in closed form for any N ∈ R. Furthermore, if N ∈ Z then the scattering data are reflectionless and so the solution q(x, t; N ) can be found more-or-less explicitly. In [17] it was noted that, with ε = c/N and τ = N t/c, the function φ(x, τ ; ε) = cq(x, t; N )/N satisfies the initial-value problem (7) iεφ τ + ε 2 2 φ xx + |φ| 2 φ = 0, φ(x, 0; ε) = c sech(x).…”

“…In this paper, we present a calculation of the scattering data for (1)- (2) analogous to the work in [19] and [21], and we also present an explicit computation of u(x, t; ε, µ) as ε ↓ 0 analogous to [17]. The asymptotic analysis of the semiclassical Cauchy problem for sine-Gordon corresponding to the work in [12,15,22,23] will be carried out in a later work.…”

Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations

“…The single-and manysoliton solutions to this equation have been extensively explored in the context of optical solitons [50][51][52][53][54][55]. We review the results pertinent to soliton dynamics in this section.…”

“…The majority of these dynamics can be understood on the basis of a simple particle model. The interaction of solitons with radiation is more mathematically involved [52,55,75] and no similarly general picture is available. In the absence of radiation, the dynamics of multiple bright solitons are dominated by the interactions and collisions between solitons.…”