Modulational Instability and Parametric Amplification Induced by Loss Dispersion in Optical Fibers Takuo Tanemura, * Yasuyuki Ozeki, and Kazuro Kikuchi Research Center for Advanced Science and Technology, University of Tokyo, 4-6-1 Komaba, Meguro-Ku, Tokyo 153-8904, Japan (Received 14 April 2004; published 13 October 2004) We show that modulational instability may arise even in the normal group-velocity dispersion regime of an optical fiber when the fiber loss (gain) varies depending on the wavelength. A simple analytical expression for the instability gain is obtained, which reveals that the odd-order terms of the loss dispersion are responsible for this phenomenon. The instability gain is measured experimentally in an optical-parametric-amplification configuration. Large parametric gain is induced in a non-phase- matched regime as we apply narrow band loss at the idler wavelength. DOI: 10.1103/PhysRevLett.93.163902 PACS numbers: 42.65.Sf, 42.65.Hw, 42.65.Yj, 42.81.Dp Modulational instability (MI) is a general feature of wave propagation in dispersive nonlinear media and is exhibited in such diverse fields as fluid dynamics [1], plasmas [2], and nonlinear optics [3–5]. In particular, MI has been widely studied in the context of optical fibers [4,5], which provide pure one-dimensional and fairly stable environments for the observation of nonlinear op- tics. When a strong continuous-wave (cw) light propagates inside an optical fiber with weak perturbations, amplitude-modulational (AM) perturbation converts to the phase-modulational (PM) perturbation through the optical Kerr effect, while the PM perturbation is trans- ferred back to the AM perturbation by the group-velocity dispersion (GVD). In the anomalous GVD regime, these effects provide positive feedback, resulting in MI, i.e., the exponential growth of the perturbations. MI can be ex- plained alternatively as the process of optical-parametric amplification (OPA), where the anomalous GVD is re- quired to satisfy the phase-matching condition among the pump carrier and two modulational sidebands [5]. In this context, observation of MI in the normal-GVD regime has been limited to special cases in which an extra phase shift is provided by an additional copropagating pump mode [6–11], by the higher-order GVD [12], or in a ring cavity configuration [13,14]. In this Letter, we show that a novel type of MI occurs in the normal-GVD regime when the fiber loss varies de- pending on the wavelength. Unlike the standard scalar MI in the anomalous GVD regime, the PM-to-AM conver- sion of the pump perturbation is induced by the loss dispersion instead of GVD. Assuming general profiles of both the loss (or gain) and GVD, we obtain a simple analytical expression for the MI gain, which indicates the explicit conditions for inducing MI in the normal- GVD regime. We then demonstrate direct observation of the loss-induced MI in an OPA configuration. Significant enhancement of the OPA gain is observed in a non-phase- matched regime of a fiber as we induce narrow band distributed loss at the idler wavelength. In fact, the closely related phenomenon of Raman- assisted OPA has been discovered by several authors [15,16], in which the Raman resonance is proved to enhance the OPA gain in a non-phase-matched regime. In addition, the recent analytical work on MI in a rare- earth-doped fiber amplifier claims that MI occurs in the normal-GVD regime when the pump is placed at the slope of the gain spectrum [17]. Our analysis reveals that the wavelength-dependent gain, induced by either Raman effect or rare-earth doping, is not necessarily required but any odd-order (asymmetrical) loss dispersion in gen- eral may cause MI in the normal-GVD regime. In the linearized regime of MI, or equivalently, under the undepleted-pump approximation of OPA, wave evolu- tion along the fiber is described by the coupled-mode equations [5]: dA 0 dz ijA 0 j 2 A 0 1 2 0 A 0 ; (1) dA dz i2jA 0 j 2 A A 2 0 A e ikz 1 2 A ; (2) dA dz i2jA 0 j 2 A A 2 0 A e ikz 1 2 A ; (3) where A 0 is the slowly varying envelope of the pump carrier, whereas A and A are those of the upper and lower sidebands of the weak modulation, respectively. 2n 2 =A eff (n 2 is the nonlinear refractive index, is the wavelength, and A eff is the effective core area of the fiber) is the nonlinear coefficient and k k k 2k 0 (k j are the propagation constants of re- spective waves) denotes the linear phase mismatch. The wavelength dependence of is ignored in the case of our interest. The additional terms j A j =2 represent the ab- sorption ( j > 0) or gain ( j < 0) experienced by respec- tive waves. We assume for simplicity. From Eq. (1), A 0 is solved as A 0 z P 0 p expi R z 0 Pz 0 dz 0 0 z=2, where Pz P 0 exp 0 z is the local pump power. By introducing new variables VOLUME 93, NUMBER 16 PHYSICAL REVIEW LETTERS week ending 15 OCTOBER 2004 163902-1 0031-9007= 04=93(16)=163902(4)$22.50 2004 The American Physical Society 163902-1