EE OUNAL OUAUM ELECRONC, O 31, O, 12, EEMER 99 22 Excess Noise Factor in Laser with Extra Cavity Devices Pawel Szczepanski, Anna Tyszka-Zawadzka, and Adam Kujawski Abstract-The efect of the nonorthogonality of the longtudinal eigenmodes in laser with an extra cavity device described by the .extra cavity distributed losse, is analyzed. An expression for the excess noise factor is derived. It is shown that the extra cavity devices change remarkably the behavior of the noise characteristics in comparison with conventional two-mirrr laser. I. NTOTION I T HAS BEEN stressed recently that the transverse eigen modes [1]-[6] and the longitudinal eigenmodes [7]-[11] of a laser resonator become nonorthogonal. Also a global efect has been investigated [12]. One major consequence of this nonorthogonality is that the coupling of the spontaneous emission fom the active medium to lasing modes is increased. Petermann [1] first derived the excess spontaneous emission factor (also called excess noise factor) which predicted an excess spontaneous emission rate or excess noise level in gain guided semiconductor lasers. Then it was realized by Haus and Kawakami [2] that such gain or loss system exhibit correlation between the noise emission into diferent propagating modes. causes that the spontaneous emission into the lasing mode has a energy geater than an energy of one extra photon per mode. Siegman [3], [4] has generalized this problem, showing that essentially the same sort of excess spontaneous emission would occur in any sort of opn-sided optical resonator or lens guide, as a consequence in essence of the nonpower orthogonal nature of the transverse modes in such structures. Excess spontaneous emission factor were calculated for the confocal unstable optical resonator [5] and for the lowest and higher tansverse modes in geometrically stable and unstable laser cavities having Gaussian variable-refectivity mirors [6]. Recently, the Siegman's teatment has been extended to incorporate also the case of nonorthogonality of the longi tudinal eigenmodes for DF [7]-[9] as well as F-P [10]-[11] lasers. For DF lasers, the problem of the nonorthogonality of the longitudinal modes has been discussed also for index- Manuscript received November 7,199a This work was supported in part by The Foundation for Polish Science. P. Szczepanski is with the Institute o Electnic Materials Technology, 0-919 Warsaw, ul. W6lczynska 133, Poland. A. Ty�zka-Zwada iththeniteoicoeleconiad oelc tronics, Wasaw University of Technology, -66 Warsaw, ul. Koszykowa 7, Poland. ^ Kujawski is with the Institute of Physics, Warsaw University of Technology, 00-66 Warsaw, ul. Koszykowa 75, Poland. IEE Log Number 94133b coupled as well as pure gain-coupled laser structure [7]-[9]. In this case, the Bragg scattering process providing optical feedback in DF structure causes simultaneously the nonuni form electric feld di�tribution and the nonorthogonality of the longitudinal eigenmodes. This leads a longitudinal excess noise factor. Moreover, the nonorthogonality of the longitudinal laser modes has been discussed [10] and verfied experimentally r11] for two-mirror laser. In this kind of laser, the nonuniform electric field distribution is observed when the gain and loss do not coincide spatially because of the point losses at the end mirors caused by the arbitrar mirror transmission. Thus, the longitudinal eigenmodes are nonorthogonal, which leads to a longitudinal excess noise factor. In paticularly, in the analysis of the longitudinal excess noise factor i n Fabry-Perot laser [10], the length of the active medium is taken to be the same as the length of the laser cavity. However in laser system often the length of the active medium is shorter than the length of the cavity and additional devices are itroduced. In general, it leads to the extra cavity distributed losses in the structure. In this paper, we extend approach presented in [10] to analyze an infuence of the extra-cavity devices on the excess noise factor in two-mirror laser. Especially, this efect can be important in a laser with saturable absorber. Because of the possible bistable operation of such a laser it may play an important role in optical signal processing and optical telecommunications systems where the noise level must be minimized. II. THORY We consid�r the laser structure shown in Fig. 1. For such a structure, the electric field of the Nth longitudinal mode can be written in Bloch notaton as follows (1) where N is the wave vector and R�Il, S�Il are the complex amplitudes of Nth counterrunning waves of the Nth longitu dinal laser modes in the frst region (i.e., in the gain medium, z c (0, L1) and in the second region with distributed losses, z c (L1,L), respectively. In our approach, similarly as in [10], we assume that the complex amplitudes are proportional to the threshold distribution. Thus, in our case we have: In the 001-919795$04`0 1995 IEEE