Mirror symmetry or parity is a fundamental symmetry in nature found on scales ranging from celestial bodies to atomic structures. It occurs when there is a mirror plane separating a shape or an object into two halves that are mirror images of each other. In the microscopic world described by quantum mechanics and in other wave phenomena, mirror symmetry warrants that the mathematical function ψ(x) describing the underlying object, such as the probability amplitude of finding an electron in a water molecule or an acoustic mode of a violin, is either symmetric or anti-symmetric about the mirror plane. In other words, the relative phase angle θ between the two halves of ψ(x) is either 0 or π. An intellectual curiosity one may have is whether there exists a “complex mirror symmetry” that would remove this restriction on θ, extending it to the entire 2π range while maintaining the identical probability density or intensity distribution of the two halves given by |ψ(x)|^2. Here we show theoretically that this complex mirror symmetry can be realized and observed in an artificial crystal such as a photonic lattice, through the inclusion of a non-Hermitian interface formed by a double-layer of optical gain and loss materials [1,2]. By utilizing complex mirror symmetry and its recursive applications, we find a straightforward paradigm to construct high-order non-Hermitian degeneracies, which can potentially increase the spontaneous emission rate and sensing sensitivity of photonic devices [3,4] by orders of magnitude.
 L. Feng, R. El-Ganainy, L. Ge, Non-Hermitian photonics based on parity-time symmetry. Nat. Photon. 11, 752–762 (2017).
 R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, D. N. Christodoulides, Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).
 W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, L. Yang, Exceptional points enhance sensing in an optical microcavity. Nature 548, 192-196 (2017).
 H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points. Nature 548, 187–191 (2017).
In photonics and quantum optics, a key challenge facing any technological application has traditionally been the mitigation of optical losses. Recent work has shown that a new class of optical materials, called Parity-Time symmetric materials, that consist of a precisely balanced distribution of loss and gain can be exploited to engineer novel functionalities for propagating and filtering electromagnetic radiation. Here we show a generic property of optical systems that feature an arbitrary distribution of loss and gain, described by non-Hermitian operators, namely that overall lossy optical systems can transiently amplify certain input signals by several orders of magnitude. We present a mathematical framework to analyze the dynamics of wave propagation in media with an arbitrary distribution of loss and gain and construct the initial conditions to engineer such non-Hermitian power amplifiers.
A study on the effects of optical gain nonuniformly distributed in one-dimensional random systems is presented. It
is demonstrated numerically that even without gain saturation and mode competition, the spatial nonuniformity
of gain can cause dramatic and complicated changes to lasing modes. Lasing modes are decomposed in terms of
the quasi modes of the passive system to monitor the changes. As the gain distribution changes gradually from
uniform to nonuniform, the amount of mode mixing increases. Furthermore, we investigate new lasing modes
created by nonuniform gain distributions. We find that new lasing modes may disappear together with existing
lasing modes, thereby causing fluctuations in the local density of lasing states.