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 ...

In this paper we consider a system of nonlinear wave equations which admits, in a linear approximation, a planewave solution with high-frequency oscillation. Then, for the wave of small but finite amplitude, we investigate how slowly varying parts of the wave such as the amplitude are modulated by nonlinear self-interactions. A stretching transformation shows that, in the lowest order of an asymptotic expansion, the original system of equations can be reduced to a tractable, single, nonlinear equation to determine the amplitude modulation.

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A perturbation method given in a previous paper of this series is applied to two physical examples, the electron plasma wave and a nonlinear Klein-Gordon equation. In these systems, and probably in most physical systems, an assumed condition for a mode of l = 0 is not valid. Consequently, the direct application of the method is impossible. In the present paper, we shall illustrate by these examples how this difficulty can be overcome to allow us to use the method. As a result we shall find that, in either case, the original equation can be reduced to the nonlinear Schrödinger equation.

The perturbation method for the nonlinear, slow modulation of a rapidly oscillatory plane wave, which was given in the first paper of this series for a class of systems of nonlinear partial differential equations, is now established for a general system of nonlinear integro-partial differential equations. It is shown that the system can be reduced to simpler nonlinear equations which in certain cases become the nonlinear Schrödinger equation. The reduction proceeds as in the first paper, and the result is then applied to nonlinear optics.

The modified C.G.L. equation which includes the effect of the finite Larmor radius of the ion is applied to the study of hydromagnetic wave propagation in a collisionless plasma. The dispersion relation is given for a linearized system in which waves propagate in the direction normal to a magnetic field. Then the dispersive effects of the finite ion Larmor radius and the electron inertia are investigated in various cases. A precursor with an oscillatory structure appears in front of the main disturbance due to the former effect, and the latter gives rise to an oscillatory trail following behind. § 1. Introduction 1The importance of the effect of the finite Larmor radius (F.L.R.) of the ions on plasma instabilities in a strong magnetic field has been discussed by many authors. 1 > They have pointed out that the difference between the Larmor radii of ion and electron leads to a difference of the mean electric fields experienced by them so that their drift velocities are different and this builds up a charge separation out of phase with the charge separation due to particle drifts such as result from the gravitational force, constant acceleration and so on. It may be expected that this situation can be found also for the propagation of hydromagnetic waves in a collisionless plasma when the characteristic wave length becomes of an order comparable to the Larmor radius of the ion. In this paper, we shall discuss the wave propagation across a magnetic field on the basis of the modified ChewGoldberger-Low equation

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