Generalized mode propagation in first-order optical systems with loss or gain

Mode propagation through first-order systems with loss or gain is derived based on a canonical transform theory developed recently. In analogy to quantum theory, creation and annihilation operators are defined for the construction of high-order modes. The evolution of these mode-generation operators through an optical system is shown to describe the propagation of the actual fields that are parameterized by two raylike labels—two guiding rays. The mode-generating operators create a set of generalized modes that may be reduced to the ordinary Her-mite–Gaussian modes only in a lossless system. Many characteristics of these generalized modes are evaluated by operator algebraic manipulations without the need for the explicit form of the actual modes that are shown also to be Hermite–Gaussian but involve complex scaling parameters.