# Unidirectional Invisibility Induced by Complex Anti-Parity-Time Symmetric Periodic Lattices
## 1 Introduction  
According to the theory of quantum mechanics, the observed physical parameters in experiments are real, and the corresponding Hamiltonian operator is Hermitian. When a light field interacts with the multi-level atomic systems, under-considering the spontaneous emission and ionization of atoms, dissipative terms are introduced, then the Hamiltonian operator to describe the system is non-Hermitian [1; 2; 3]. The eigenvalues of non-Hermitian Hamiltonian are usually complex containing imaginary parts. However, the non-Hamiltonian that satisfies the parity-time (PT) symmetry has a real spectrum, which extends quantum mechanics into a new research field [4; 5; 6; 7; 8; 9; 10]. PT-symmetry and non-Hermitian optics are new concepts and have become a research hotspot in recent years [11; 12; 13; 14; 15; 16; 17; 18; 19]. It is worth noting that the study of Hermitian and non-Hermitian properties are not limited to electronic systems, but also can be extended to the field of optics [20; 21]. Although the Schrodinger equation describing a single particle is different from the Maxwell equation describing electromagnetic field in physical origin, the electromagnetic field propagation equation is similar to the Schrodinger equation in mathematical form under the slowly varying envelope approximation [22; 23]. This similarity provides the possibility of studying PT symmetry and non-Hermiticity in optical systems. Therefore optics has the natural advantage of being used to explore all sorts of non-Hermitian predictions that used to be considered difficult in quantum systems [24; 25; 26]. The introduction of gain and loss in photonic crystals, as well asthe clever design of various structures, can improve the luminous efficiency of materials and obtain fascinating nonlinear and topological properties [27; 28; 29; 30]. The propagation constant in PT symmetric phase is a real number. Although there is gain and loss, the energy of the system is conserved. When the gain/loss ratio \(g\) is much smaller than the coupling coefficient, it corresponds to the PT symmetric phase, and the PT symmetric supermodel appears in the system [21]. For the larger \(g\), the PT phase will be broken. At this time, severe mode mismatch will occur in the two waveguides, and the energy coupling of the mode field is suppressed, resulting in the occurrence of asymmetric light intensity distribution [31]. In the vicinity of spontaneous PT-symmetry breaking point, the PT symmetric periodic structure medium shows unidirectional invisibility [32]. In a wide frequency range near the Bragg point, when light enters from one end, the outgoing light passes through unaffected. But when light comes in from the other end, the reflected light is enhanced. The anti-PT symmetry which arises from the charge-conjugation symmetry has also induced many fascinating phenomena, such as flat broadband light transport [33], topological bound modes [34] and chiral mode conversion [35]. The refractive index of material obeys \(n(z)=n^{\star}(-z)\) and \(n(z)=-n^{\star}(-z)\) in PT and anti-PT symmetric systems [16; 17; 36], respectively. If expressed in terms of dielectric constant, PT and anti-PT symmetric systems both satisfy the condition \(\varepsilon(z)=\varepsilon^{\star}(-z)\). Further, we hypothesized whether new optical properties would be obtained when \(\varepsilon(z)=-\varepsilon^{\star}(-z)\). At this point, the refractive index satisfies \(n(z)=-in^{\star}(-z)\), which is called complex anti-PT symmetry here. This new symmetry can be seen as a combination of anti-PT symmetry and rotation of \(\pi\)/2. It further extends the scope of symmetry and may lead to more novel optical properties. In this work, we investigated the unidirectional invisibility in the complex anti-PT-symmetric periodic lattices. We showed that the reflection was anisotropic as light impinged upon the lattices from the left and right end.