Eigenvalue dynamics in the presence of nonuniform gain and loss

Abstract

Loss-induced transmission in waveguides, and reversed pump dependence in lasers, are two prominent examples of counter-intuitive effects in non-Hermitian systems with patterned gain and loss. By analyzing the eigenvalue dynamics of complex symmetric matrices when a system parameter is varied, we introduce a general set of theoretical conditions for these two effects. We show that these effects arise in any irreducible system where the gain or loss is added to a subset of the elements of the system, without the need for parity-time symmetry or for the system to be near an exceptional point. These results are confirmed using full-wave numerical simulations. The conditions presented here vastly expand the design space for observing these effects. We also show that a similarly broad class of systems exhibit a loss-induced narrowing of the density of states.Recently, the study of parity-time (PT ) symmetric optical systems has highlighted the importance of exploring non-Hermitian systems with patterned gain and loss [1][2][3][4][5][6][7][8][9][10][11], and has led to the discovery of a remarkable array of phenomena, such as loss-induced transmission in waveguides [12], unidirectional transport behavior [13][14][15][16][17], reversed pump dependence in lasers [18][19][20], and band flattening in periodic structures [4,[21][22][23][24]. These effects are leading to new possibilities for constructing on-chip integrated photonic circuits for the manipulation of light.Here, we focus on two of these effects: loss-induced transmission in waveguides, and reversed pump dependence in lasers. Both of these effects are fascinating since they are quite counter-intuitive. They are also of potential practical importance in providing novel mechanisms for optical switching based on gain or loss modulation. In loss-induced transmission, loss is added to one of two otherwise identical, parallel coupled waveguides [12]. After a critical amount of loss is added, further increases in the absorption also increases the total transmission through the waveguide pair. Likewise, reversed pump dependence can be observed in laser systems consisting of two coupled cavities [18][19][20]25]. First, one of these cavities is pumped such that the total system begins to lase. However, if the pump in the first cavity is then held constant and the gain in the second cavity is increased, the total output lasing power can be seen to decrease until the total gain distribution becomes relatively uniform, so long as the added gain is sufficiently greater than the losses of the unpumped system. If the laser is close to threshold after gain is added to the first cavity, this mechanism can drive the laser below threshold.Initially, these types of counter-intuitive effects were found in PT symmetric systems in their broken phase, and thus the onset of these behaviors was associated with the occurrence of an exceptional point [12]. Subsequent analyses demonstrated that loss-induced transmission in waveguides and reverse pump dependence in lasers could be found in s...

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