PHYSICAL REVIEW A VOLUME 32, NUMBER 2 Semiclassical theory of coupled lasers AUGUST 1985 Sami A. Shakir and Weng W. Chow Institute for Modern Optics, Department of Physics and Astronomy, University of ¹w Mexico, Albuquerque, ¹av Mexico 87131 (Received 10 October 1984) The semiclassical equations of motion for a system of coupled lasers are developed and the fre- quency locking of the lasers comprising the system is analyzed. It is shown that the frequency- coupling range, in terms of the coupled cavities mismatch, is proportional to the coup1ing coeffi- cient. For a system where the cavities are uniformally filled with the active medium, the coupling vanishes regardless of the transmittance of the coupling mirrors. Our theory is valid for a11 values of coupling and for any number of lasers in the array. It may also be adapted to study different types of coupling arrangements. I. INTRODUCTION The use of two consecutive Fabry-Perot interferometers for the purpose of better axial-mode selectivity was sug- gested by Kleinman and Kisliuk' and has since been fur- ther investigated by several workers. Coupling two or more resonators can lead to desirable properties such as selective mode suppression, enhancement of intracavity laser power, frequency stabilization against cavity-length fluctuations and phased laser arrays. Spencer and Lamb had previously studied the coupled-laser problem. Their treatment, which was based on expanding the laser field in terms of the individual passive-resonator eigenmodes is hmited to the weak- coupling regime. In order to study the strongly-coupled- laser problem, one has to expand the laser field in terms of composite passive-resonator eigenmodes. Recently, we have shown that by using the composite resonator eigenmodes, the semiclassical equations for the fields of coupled lasers are very similar in form to those of a single-resonator laser. This is a useful result because the solution of the single-resonator problem is well known. 'o The effects of the coupling between lasers are primarily due to changes in the passive-resonator-mode structure. In particular, the frequency spacings between the modes are no longer uniform, and the cavity losses of different modes are different. In this paper we derive the semiclassical equations of motion for a system of coupled lasers. We apply these equations to study frequency locking in coupled lasers. The ability to lock the frequencies of the coupled lasers in the presence of resonator-length fluctuations is important for phased laser-array applications. The approach adopt- ed here is general in that it places no restriction on the strength of coupling or the relative lengths of the coupled lasers. It also applies to a variety of geometrical forms of coupling. Throughout this paper, homogeneous broaden- ing is assumed; however, it is a straightforward problem to derive the corresponding equations for an inhomogene- ously broadened active medium. Our theory is based on the knowledge of the composite resonator passive modes. In Sec. II, a simple method is developed for computing the eigenfunctions and eigenfre- quencies of these modes. Even though we developed the analysis for resonators coupled in series, the same ap- proach applies to other forms of coupling (Sec. VII). In Sec. III, the results are applied to the case of two coupled resonators. The semiclassical equations of motion are de- rived in Sec. IV. Frequency locking is discussed in Sec. V where the cou- pled resonator field equations are solved numerically for the stable solutions. In Sec. VI the field equations are re- duced to equations similar to those encountered in con- nection with ring lasers. The "decoupled" approxima- tion' is used to solve these equations. In Sec. VII the methods of Sec. II are applied to a different form of cou- pling (star coupling). II. THE COMPOSITE RESONATOR EIGENMODES Consider a high-Q composite resonator consisting of M subresonators (see Fig. 1). These subresonators are cou- pled via transmitting mirrors. These mirrors are charac- terized by their reflectance amplitudes r (j), r(j), and the transmittance amplitude t(j) Here, r. (j) [r(j)] corre- sponds to the reflectance amplitudes when a mirror is viewed from the left (right). The transtnittance ampli- tudes as viewed from both sides are taken to be equal since we are assuming that the refractive index of the medium on both sides of the mirrors are the same. The choice of r (j), r(j), and t (j) is not completely arbitrary. In the case of a lossless mirror, they must satisfy the rela- tion r(j )Itj () = r*j ()It'(j). — The intracavity electric field is analyzed in terms of the right and left running waves, A+(j) and A (j), respec- tively. These running waves have constant, but different amplitudes in each (passive) subresonator. To relate the wave amplitudes of these waves in different sections of the composite resonator, we first relate the waves at both sides of a given mirror, say mirror j which is located at z =zj. The waves to the left are labeled A+(j) and A (j), while the ones on the right are labeled A +(j) and A (j). Here, A(j) is a shorthand for A(zJ). With the aid of Fig. 2, one can immediately write the following re- lations: 32 . 983 198S The American Physical Society