An exact result for the spectral density of intensity variations that occur after propagation of ergodic light in a medium having lowest-order-only group-velocity dispersion is obtained and applied to the problem of semiconductor laser phase noise to intensity noise conversion in a single-mode optical f iber. It is shown that the intensity spectrum after propagation formally approaches, for a large laser linewidth or a long (or highdispersion) f iber, the intensity spectrum of a thermal source having the same line shape as the laser. 2000 Optical Society of America OCIS codes: 060.2430, 140.5960, 260.2030, 270.2500 The conversion of optical phase variations to intensity variations by group-velocity dispersion (GVD) in optical fiber is a well-known phenomenon which occurs both for phase modulation and for phase noise. The effect of GVD in fiber on variations due to sinusoidal phase modulation can be modeled by expanding the time dependence of the phase-modulated input field in a Bessel-coeff icient Fourier expansion.
1Phase noise can also be considered following such an approach, but only with the approximation that phase variations at a given frequency contribute solely to intensity variations at that same frequency -otherwise the calculation becomes intractable.
2Analyses valid in the limiting cases of small amplitude variations (i.e., a small-signal model 3 ) and small propagation distance and low frequency 4 have also been presented. The small-signal model assumes that the optical field can be described as E͑t͒ p I 0 ͓1 1 D͑t͒ 1 if͑t͔͒exp͑iv 0 t͒, where D͑t͒ and f͑t͒ are small, zeromean amplitude variations and phase variations, respectively. This leads to an expression for the relative intensity noise (RIN) after dispersive propagation of the form RIN͑V, z͒ 4͓S DD ͑V͒ cos 2 u 2 Re S Df ͑V͒ sin 2uwhere V is the angular frequency, z is the propagation distance, and for lowest-order GVD, u 2b 00 0 zV 2 ͞2. Here S DD ͑V͒ is the spectral density of the normalized amplitude variations of the source, S ff ͑V͒ is the spectral density of phase variations of the source, and S Df ͑V͒ is the cross-spectral density characterizing amplitude-phase correlations of the source.The above-described model assumes that the total phase excursion is small, jf͑t͒j ,, 1. For a laser, whose phase exhibits a nonstationary random walk, this is true only over an interval on the order of the coherence time, t coh . The small-propagation-distance model 4 does not require that phase variations be small but yields results that are valid only over a narrow range of (low) frequencies. Practically speaking, both these approaches can fail in cases of large field linewidths and in situations involving large-angle phase modulation. The former is the subject of this Letter; the latter, which involves nonergodic fields, will be addressed in a separate publication.
5In this Letter we present an exact theory for the spectral density of the intensity of ergodic (i.e., time average ensemble average) light after propagation in a medium exhibiting lowe...