Exceptional points (EPs), which are branch point singularities of the spectrum, associated with the coalescence of both eigenvalues and corresponding eigenstates, lead to interesting phenomena, such as level repulsion and crossing, bifurcation, chaos, and phase transitions in open quantum systems described by non-Hermitian Hamiltonians , , , . EPs have been studied in lasers , coupled dissipative dynamical systems , mechanics , electronic circuits , gyrokinetics of plasmas , and atomic as well as molecular systems . In the past few years, EPs in non-Hermitian parity-time (PT)-symmetric systems have attracted considerable attention . In quantum mechanics, non-Hermitian Hamiltonians may still possess entirely real energy spectra as long as they respect PT-symmetry. In general, the Hamiltonian H=p2/2+V(r), associated with a complex potential V(r), is PT-symmetric as long as the complex potential satisfies the condition V(r)=V*(−r) . However, once a non-Hermiticity parameter exceeds a certain threshold, the eigenvalues of the Hamiltonian cease to be all real. This threshold is associated with the appearance of the EP, where the eigenvalue branches merge and PT-symmetry breaks down. Based on the close analogy between the Schrödinger equation in quantum mechanics and the wave equation in optics, PT-symmetry in optics requires that n(r)=n*(−r), which implies that the real part of the refractive index n(r) should be an even function of position, whereas the imaginary part must be an odd function , .
The constructed PT-symmetric optical structures with balanced gain and loss can lead to a range of extraordinary phenomena, including novel beam refraction , , power oscillation , , loss-induced transparency , nonreciprocal nonlinear light transmission , , , perfect absorption , , , teleportation , optical switching , optical tunneling , mode conversion , super scattering , and various other novel nonlinear effects , , . In addition, there has been significant progress in using PT-symmetric periodic optical structures to attain unidirectional light reflectionlessness , . In such structures, the reflection is zero when measured from one end of the structure at optical EPs and nonzero when measured from the other end.
Unidirectional reflectionlessness at EPs can be understood by considering a general two-port optical scattering system, as shown in Figure 1. The optical properties of this system can be described by the scattering matrix S defined by 
where HL+ and HR+ are the complex magnetic field amplitudes of the incoming modes at the left and right ports, respectively. Similarly, HL− and HR− are the complex magnetic field amplitudes of the outgoing modes from the left and right ports, respectively. In addition, t is the complex transmission coefficient, whereas rf and rb are the complex reflection coefficients for light incidence from the left (forward direction) and from the right (backward direction), respectively. We note that, because PT-symmetric structures are in general reciprocal, the transmission coefficients in both directions must be the same . However, the reflection coefficients from the left and right directions are not constrained by reciprocity and can therefore be different. We also note that the scattering matrix S as defined here [Equation (1)], which is in general nonsymmetrical, is different from the typical convention used in electromagnetics . In general, the matrix S is non-Hermitian, and its corresponding complex eigenvalues are Its eigenstates, which are for rb≠0, are not orthogonal. The two eigenvalues and the corresponding eigenstates can be coalesced and form EPs. Such non-Hermitian degeneracies represent scattering states with unidirectional reflectionless propagation in the forward (rf=0, rb≠0) or backward (rb=0, rf≠0) direction.
In the case of unidirectional reflectionless propagation in the forward direction (rf=0, rb≠0), the scattering matrix S eigenvalues coalesce into λc=t and the eigenstates Ψ± coalesce into the only eigenstate Ψc=(1,0)T. Because the scattering matrix eigenstates coalesce into the only eigenstate Ψc=(1,0)T, they no longer form a complete basis. We note that the eigenstate Ψc corresponds, through Equation (1), to a well-defined physical scattering state with (HL+, HR+)=Ψc=(1,0)T and (HL−, HR−)=λcΨc=(t,0)T. In other words, the eigenstate corresponds to a state with unidirectional reflectionless propagation for light incident from the left. A similar discussion holds for the case of unidirectional reflectionless propagation in the backward direction (rb=0, rf≠0).
The two complex eigenvalue solutions of the scattering matrix S construct a multivalued Riemann surface , . They are on two branches of the Riemann surface. The two branches, represent two superposition states between forward and backward light scatterings, including both transmission and reflection. The solutions of two branches coalesce and lead to EPs, corresponding to singularity points in the complex Riemann surface . The reflection in both directions is due to the superposition of such two solutions . At the EPs, light scattering is a destructive interference of two branch solutions. Therefore, the corresponding modal interference between two branch solutions at the EPs suppresses the reflection in one direction but not the other . EPs exist in a larger family of non-Hermitian Hamiltonians . Unidirectional reflectionless propagation can also be observed in non-PT-symmetric optical systems with unbalanced gain and loss .
In this paper, we review the fundamental physics and the latest developments on unidirectional reflectionless light propagation at EPs. In Section 2, we first review the unidirectional reflectionless propagation at EPs in PT-symmetric optical systems. Section 3 is devoted to reviewing unidirectional reflectionless propagation at EPs in non-PT-symmetric optical systems. Finally, in Section 4, we conclude this review and offer our perspectives on possible future directions in this field.
2 Unidirectional reflectionless propagation in PT-symmetric systems
To realize a PT-symmetric optical structure with unidirectional invisibility, Lin et al. used coupled mode theory (CMT) to theoretically investigate a 1D optical structure consisting of a periodic grating with refractive index distribution along the propagation direction n(z)=n0+n1cos (2βz)+in2sin(2βz) for |z|<L/2 (Figure 2A) . This grating is embedded in a homogeneous medium having a uniform refractive index n0 for |z|>L/2 (Figure 2A). Here, β is the grating wave number. The terms n1cos(2βz) and n2sin(2βz) correspond to the periodic distribution of the real and imaginary parts (gain and loss) of the complex refractive index in the grating, respectively. For PT-symmetric optical systems, the eigenvalues of the S matrix (Figure 1) must be either a unimodular or a nonunimodular inverse conjugate pair . In the former case, the system is in the exact PT phase, whereas, in the latter one, it is in the broken symmetry phase , , . For the complex periodic structure considered in Figure 2A, the transition from one phase to another takes place when n1=n2. At n1=n2, the eigenvalues of the S matrix and their corresponding eigenvectors coalesce and form an EP, indicating that the phase transition takes place exactly at this EP. In fact, in quantum mechanics, the phase transition due to a non-Hermitian degeneracy is another phenomenon associated with EPs , .
Figure 2B shows the transmission and reflection coefficients as a function of the detuning δ=β−k, where k=ωn0/c. At n1=n2 (EP), the reflection in the left direction RL=|rf|2 is zero, whereas the reflection in the right direction RR=|rb|2 is nonzero. Meanwhile, the transmission (T=|t|2) is unity, which implies that the Bragg scatterer is invisible when the light is incident from left (light propagating as if the Bragg scatterer is absent). Note that the transmitted wave cannot be detected from interference measurements, as the phase ϕt of the transmission coefficient t is zero at n1=n2 (EP). In addition, the transmission delay time τt=dϕt/dk is zero at n1=n2, which indicates that the time of flight of the light transmitted through a certain distance is the same whether there is a Bragg scatterer or not. Therefore, reflectionlessness in the proposed PT-symmetric Bragg scatterer results in invisibility. It is also clear that the reflection coefficients are symmetric, that is, RL=RR, when no gain or loss is included in the system (n2=0). In addition, the generalized power representation relates all the elements in the scattering matrix S and is essentially the power summation of the superposition of the two eigenstates [Equation (1)] . Note that the reflection of an incident wave from the left side of the structure is subunitary, whereas the reflection from the right side is superunitary. Therefore, T+RR in this system (Figure 2A) can be larger than one (Figure 2B) . When the reflection in the forward and backward directions are equal, we obtain which is the power conservation relation for an optical system without gain or loss (Figure 2B).
Later on, Longhi reconsidered the scattering properties of this sinusoidal PT-symmetric Bragg scatterer (Figure 2A) using modified Bessel functions of the first kind . His analytical results show that the unidirectional reflectionlessness only occurs for Bragg scatterers with short length L at EPs and breaks down for extremely long length scatterers. Note that the derivation of coupled-mode equations in Ref.  was based on multiscale asymptotic techniques and the rotating wave approximation . Subsequently, Sarisaman demonstrated unidirectional reflectionlessness and invisibility in the TE and TM modes of a PT-symmetric slab system consisting of a separated pair of balanced gain and loss layers with a gap . Kalish et al. reported that one-way invisibility could be obtained in randomly layered optical media with PT-symmetric refractive index . Midya designed unidirectionally invisible complex optical crystals with balanced gain and loss based on nonrelativistic supersymmetry transformations . Fu et al. demonstrated the existence of two EPs in a waveguide system consisting of zero-index materials with PT symmetry, which could induce unidirectional transparency . Rivolta and Maes observed unidirectional visibility in a structure consisting of a waveguide and a finite chain of side-coupled resonators . Fleury et al. constructed a unidirectional reflectionless optical system using two metasurfaces characterized by a PT-symmetric impedance distribution .
Regensburger et al. performed scattering experiments on a periodic PT-symmetric temporal Bragg scatterer by imposing a periodic phase modulation only within a finite time window . Outside this time window, the periodic potential does not have an effect on light traveling, and the Bragg scatterer reflects light coming from both sides of the scatterer. When gain and loss are added to the phase modulation in a PT-symmetric fashion within the time window, unidirectional invisibility occurs at EPs. Hahn et al. first observed totally asymmetric diffraction in a PT-symmetric photonic lattice at EPs using vector-field holographic interference of two elliptically polarized pump beams on azobenzene-doped polymer thin films .
PT-symmetry in surface plasmon polaritons (SPPs) has been demonstrated by Yang and Mei . SPPs, which are surface waves propagating along the interface between a metal and a dielectric , can concentrate electromagnetic energy at volumes of subwavelength scale and enable the manipulation of light beyond the diffraction limit , , , , , , , , , . Yang and Mei introduced a periodic PT-symmetric modulation on the effective refractive index of a metal-dielectric waveguide in the cylindrical coordinate system. After a coordinate transformation, they realized a 3D cylindrical unidirectional PT-cloak at EPs for SPPs. Zhu et al. have also demonstrated a one-way invisible cloak based on a transformed PT-symmetric optical potential at EPs .
Unidirectional reflectionlessness can also be attained in optical coupled-resonator systems at EPs , . Jin et al. studied the scattering of rhombic ring form coupled resonators with enclosed synthetic magnetic flux . The scattering center is a two-arm Aharonov-Bohm interferometer. The magnetic flux induces nonreciprocal tunneling phase. In the presence of balanced gain and loss, the rhombic ring structure is under reflection PT-symmetry, which induces asymmetric reflection and reciprocal transmission. The optical gain and loss in the coupled-resonator optical systems can be experimentally realized using InGaAsP quantum wells and chrome or graphene layers, respectively, on top of the resonator , , , , .
Feng et al. have demonstrated an approach that leads to unidirectional reflectionless light propagation in a microscale silicon-on-insulator (SOI) waveguide platform without using PT-symmetric structures with balanced gain and loss . They used only purely passive materials without gain, which relaxes the requirements of the fabrication, as optical gain is difficult to achieve using conventional complementary metal-oxide-semiconductor silicon technology. As shown in Figure 3A, they introduced a periodic passive PT-symmetric modulation of the dielectric permittivity along the direction of propagation Δε=cos(qz)−iδsin(qz) [instead of ε=cos(qz)−iδsin(qz)] into the SOI waveguides. Here, δ controls the relative strength and phase between the sinusoidal and cosinusoidal modulations, and q is the wave vector of the fundamental mode at the wavelength of 1550 nm.
The eigenvalues of the S matrix of the structure in Figure 3A are both unimodular but with an additional attenuation term included in the system for 0<δ<1. For δ>1, the eigenvalues are a nonunimodular inverse conjugate pair with the same attenuation term. Therefore, δ=1 is the phase transition point corresponding to the EP, where both eigenvalues and eigenvectors become degenerate. Figure 3B and C shows that, at δ=1, the reflection in the backward direction (Rb) is zero, whereas the reflection in the forward direction (Rf) approaches ~40% as the modulation length L increases.
For practical implementation, periodically arranged sinusoidal-shaped combo structures are applied on top of a Si waveguide embedded inside SiO2 to mimic the passive PT modulation on a macroscopic scale, in which imaginary part modulation is implemented with 14 nm Ge/24 nm Cr structures and 51 nm Si layers are used for real part modulation (Figure 3D). Figure 3E shows the fabricated passive PT-symmetric waveguide. The sinusoidal-shaped combo structures are first patterned in polymethyl methacrylate (PMMA) by electron beam lithography followed by evaporation and lift-off of Si and Ge/Cr in two steps. After these steps, the Si waveguide is formed by electron beam lithography and dry etching. The measured reflection spectra of the fabricated passive waveguide are in agreement with the theoretical predictions (Figure 3F). Asymmetric light propagation in this passive system is associated with the fact that EPs exist in a larger family of non-Hermitian Hamiltonians . Using a similar method, Feng et al. proposed a passive PT-symmetric metawaveguide on an SOI platform for asymmetric interferometric light-light switching with a weak control beam, which enables coherent perfect absorption of a strong signal beam, based on the asymmetric reflection of the metawaveguide near the EP . The high ratio between the strong signal beam and the weak control beam is in contrast to previous approaches of controlling light in all-optical active devices , , , , , ,  and is promising for low-power next-generation optical networks.
In addition to the approach described above to achieve passive PT-symmetric SOI waveguides on a microscale platform, Jia et al. introduced a simple route to implement passive PT symmetry in the modal effective index of large-area (~cm2) organic thin-film waveguides . They used interference lithography to write a sinusoidal grating profile into a thin film of photoresist and deposit high extinction coefficient blue pigment copper phthalocyanine (CuPc) to coat the “windward” grating facets (Figure 4A). This arrangement results in a modal effective index with a passive PT-symmetric profile of the form (z)≈(Δneff/2)[1+cos(qz)]+i(Δkeff/2)[1−sin(qz)] for the fundamental guided mode. Figure 4B shows scanning electron micrographs (SEMs) of a fabricated composite waveguide before (top panel) and after (bottom panel) planarization of a top photoresist layer.
The optical property of the passive PT-symmetric waveguide is investigated using Kretschmann-coupled diffraction in Littrow : first, the incident laser beam is evanescently coupled into the fundamental TE mode of the composite waveguide; then, the reverse-going waveguide mode is generated by grating reflection and subsequently evanescently coupled out to air (Figure 4C). Thus, measuring the diffraction efficiency, defined as |Edif/Einc|2, for left and right incident light provides a proxy probe of the forward- and backward-going modal reflectivity within the waveguide. Here, Edif and Einc are the incident plane wave and diffractive wave amplitudes, respectively. At EPs, light in-coupled into the forward-going (left to right) mode can scatter into the backward-going (right to left) mode, giving rise to Littrow diffraction but not vice versa (Figure 4D).
As shown in Figure 4E, when δ=Δkeff/Δneff=0 (neither gain nor loss is included in the waveguide layer), the measured diffraction efficiency as a function of incidence angle for a TE-polarized 640 nm probe beam shows that the backward and forward Littrow diffraction efficiencies are similar. In contrast, at the EP, where δ=1, the backward Littrow diffraction efficiency is almost suppressed (Figure 4F). The reflectivity in the backward and forward directions is equal even in the presence of gain or loss. This is similar to the transmission in the structure of Figure 2B, which is the same in both directions regardless of the presence of gain or loss (Figure 2B). In addition, Yan and Giebink exploited vapor-deposited organic small molecules as a platform to realize passive PT-symmetry breaking and demonstrated unidirectional reflectionlessness at EPs in a composite organic thin film via complex refractive index modulation . Zhu et al. designed a passive PT-symmetric grating with asymmetric diffraction arising from the spontaneous PT-symmetry breaking at EPs in a wide range of incidence angles .
The use of gain media in optical waveguides can lead to active optical devices and to material loss compensation , , , , . Hahn et al. investigated the unidirectional reflectionlessness at EPs in PT-symmetric gratings based on active dielectric-loaded long-range SPP (DL-LRSPP) waveguides . To obtain a PT-symmetric profile, they designed a step-in-width grating structure consisting of a thin Ag stripe on an active polymer bottom cladding with an active polymer ridge (Figure 5A). The polymer used is PMMA doped with IR140 dye.
Normally, in a typical waveguide, it is impossible to achieve a PT-symmetric profile because both the real (nR) and the imaginary (nI) parts of the effective index change monotonically with the geometric parameters. Thanks to the coupling between the fundamental mode and an asymmetric high-order mode of the waveguide, when the ridge width is within specific ranges, the effective index of the DL-LRSPP waveguide has the potential to achieve a PT-symmetric profile. Figure 5B shows the effective index of the DL-LRSPP waveguide without gain as a function of the ridge width for a 35-nm-thick Ag stripe at a wavelength of 880 nm. One observes a large dip in the imaginary part of the effect index nI and a peak then a dip in the real part of the effect index nR. By arranging four widths within one step-in-width grating period in the sequence w3, w2, w4, w1, a passive PT-symmetric profile is obtained (Figure 5A and C). The thickness of the Ag stripe tag=35 nm was chosen to have ΔnR=ΔnI, which corresponds to an EP. Figure 5D shows the unidirectional reflection light propagation in such a PT-symmetric grating at its EP. When the gain of IR140-doped PMMA is included to make the average value of nI zero or positive, the maximum reflection in the left direction (Rl) is greatly enhanced, and the grating can even become an active PT-symmetric grating. Moreover, by placing a gain waveguide within a cavity formed by concatenating two gratings with opposite nonreflective ends, a high-performance laser can be created at EPs .
In all cases presented above, to achieve PT symmetry, the dielectric permittivity profile is chosen so that ε(z)=ε∗(−z), whereas the magnetic permeability is equal to the one of free space (μ=μ0). If, however, magnetic materials (μ≠μ0) are considered, a similar requirement must be imposed on the magnetic permeability, that is, μ(z)=μ∗(−z), to satisfy the PT symmetry condition . Interestingly, Li et al. explored the possibility of cross-matching ε(z) to μ∗(−z), namely, ε(z)=μ∗(−z), to establish PT symmetry . More specifically, they investigated a 1D photonic crystal (PhC) consisting of alternating purely dielectric slabs with ε1=2.25+iγ, μ1=1 and purely magnetic slabs with μ2=2.25−iγ, ε2=1 (Figure 6A). These two types of media have matched gain and loss coefficients. Note that so that the spatial PT-symmetry still holds.
Figure 6B and C shows the forward and backward reflection spectra, respectively, as a function of γ. The dips (red color in Figure 6B), reaching a value of zero, correspond to unidirectional reflection. To provide further insight into the unidirectional character in reflection, the eigenvalue behavior of constitutive matrix C−1 against γ is shown in Figure 6D. The optical property of a single unit cell can be described by the constitutive matrix C−1
where Ex1, Hy1 are the fields on the left side of the cell, Ex2, Hy2 are the fields on the right side, and a is the lattice constant. The constitutive matrix C−1 can be written in terms of the S matrix [Equation (1)] of a single unit cell through a bilinear transformation
Because unidirectional reflectionlessness occurs at an EP of the scattering matrix S, we can equivalently study the eigenvalue behavior of the matrix C−1. The eigenvalues of matrix C−1 are pairs of two purely real numbers for γ<0.107, coincide at an EP for γ=0.107, and then split into a complex conjugate pair (Figure 6D). This eigenvalue behavior is equivalent to the transition of PT-symmetric Hamiltonians in quantum mechanics from the PT-symmetric phase to the PT-broken phase , , , . The EP of the constitutive matrix C−1 is actually the EP of the S matrix, which leads to the unidirectional reflectionless behavior shown in Figure 6B and C.
When more material potentials, such as the magnetic permeability, are considered, additional degrees of freedom exist in establishing PT symmetry. An antisymmetric PT-photonic structure under combined PT operations, i.e. n(−z)=−n∗(z), with balanced positive- and negative-index materials, has been reported, where the magnetic permeability was required to satisfy μ(−z)=−μ(z) . Studies showed that light propagation in optical lattices of driven cold atoms with PT-antisymmetric susceptibilities, i.e. χ(z)=−χ∗(−z), exhibited EPs (also known as non-Hermitian degeneracies), at which complete unidirectional reflectionless light propagation was observed , .
In addition, a few recent studies of multidimensional PT-symmetric systems have demonstrated intriguing properties not found in 1D PT-symmetric systems, such as conical diffraction, third-order EPs, continuous rings of EPs, and rotating input by asymmetric coupling between wave vectors , , , , , , . Most recently, Fan et al. considered 2D PT-symmetric PhCs, whose non-Hermitian primitive cell is an integer multiple of the primitive cell of the underlying Hermitian system . Similar to many other low-dimensional PT systems, PT-symmetric PhCs can also exhibit unidirectional reflection behavior when light comes from the left and right sides of PhCs.
3 Unidirectional reflectionless propagation in non-PT-symmetric systems
As we have mentioned in the previous sections, there is a large family of non-Hermitian Hamiltonians that can produce merged branches of eigensolutions through accidental degeneracy . Therefore, can a non-PT-symmetric system, in which the relation n(−z)=n∗(z) for the refractive index profile does not hold, lead to asymmetric reflection? It has been reported that unidirectional reflectionlessness can be realized in a two-layer non-PT-symmetric slab structure by tuning both the real and imaginary parts of the refractive index of each layer at EPs .
Moreover, Horsley et al. completely suppressed scattering in one direction in a planar inhomogeneous dielectric structure, in which the spatial distributions of the real and imaginary parts of the dielectric permittivity are related by Kramers-Kronig relations . Specifically, they considered a monochromatic electromagnetic wave propagating in the x–y plane within a medium with a positive background contribution εb plus a spatially varying part α in the dielectric permittivity ε: ε(x)=εb+α(x) (Figure 7A).
If we expand the scattered field induced by the variation α(x) as a series, and solve the Helmholtz equation for the TE polarization, the first and nth terms in this series can be expressed as
where is the spatial Fourier transform of α(x), G(k) is the retarded Green function, and ky determines the angle of incidence. If (k<0)=0, then the first item is composed of only right-going waves for any value of K or, in other words, for any angle of incidence. Based on Equation (5), every successive term also contains only right-going waves if is composed of right-going waves and (k<0)=0. Interestingly, the requirement (k<0)=0 can be satisfied when the permittivity profile α(x) is holomorphic (analytic) in the upper half complex plane, i.e. Im(x)≥0, and is of Kramers-Kronig type :
where P is the principal part of the integral. We consider a specific example with The overall permittivity profile of the medium is then
which is plotted in Figure 7B. Here, ξ and A set the spatial scale and amplitude of the profile, respectively.
The one-way scattering behavior is confirmed in Figure 7C and D, which show electric field profiles for a point source placed at either side of x=0. The absence of reflection is clearly observed when light is incident from the left. Figure 7E and F shows the electric field when only the real (Figure 7E) or the imaginary (Figure 7F) parts of the permittivity profile [Equation (7)] are considered. In fact, if the imaginary part of α(x) is set to be symmetric about x=0, then the real part calculated based on Equation (6) is antisymmetric and vice versa. Thus, the spatial Kramers-Kronig relations can generate permittivity profiles that exhibit PT-symmetry [α(−x)=α*(x)]. In other words, compared to the specific class of PT-symmetric complex profiles, the permittivity profiles associated with Kramers-Kronig relations are more general nonreflecting profiles, which can achieve unidirectional reflectionlessness at EPs.
The refractive index modulation profiles that we have considered so far for synthesizing PT-symmetric or non-PT-symmetric optical structures require careful tuning of both the real and imaginary parts of the refractive index of the constituent materials, which increases the difficulty in practical realization. Therefore, the question arises as to whether it is possible to achieve asymmetric reflection in a non-PT-symmetric system by tuning only the real or only the imaginary part of the refractive index of the material. Feng et al. recently demonstrated unidirectional reflectionless light transport at EPs in a conventional large-sized nonperiodic multilayer structure, which was fabricated by alternating thin film depositions of lossy amorphous silicon and lossless silica layers on a cleaned glass wafer using plasma-enhanced chemical vapor deposition (43 nm silica/9 nm silicon/26 nm silica/23 nm silicon; Figure 8A) . The refractive index of silica at the wavelength of interest is 1.46, and the complex refractive index of amorphous silicon is 4.86+iγ, so that the proposed structure is clearly non-PT-symmetric. Using the transfer matrix theory , one can show that, by modulating only the imaginary part of the refractive index of amorphous silicon, the eigenvalues of the S matrix of this multilayer system coalesce and form an EP when γ=−0.65. The reflection spectra of the structure for both forward and backward directions were numerically calculated and experimentally measured, as shown in Figure 8B and C, respectively. Unidirectional reflectionless light transport is clearly observed around the wavelength of 520 nm, where the reflection in the forward direction is significantly suppressed due to the existence of the EP.
In the presence of loss, this optical system is analogous to open quantum systems that are subjected to dissipation and characterized by complex non-Hermitian Hamiltonians. Thus, the existence of EPs in this purely lossy non-PT-symmetric optical system also provides an opportunity to control the unidirectional reflection of light. Because of the associated unidirectional reflectionless light propagation at the EP, this large-sized non-PT-symmetric multilayer structure can form imaging in reflection under sunlight illumination only from the backward direction. An image formed in the backward direction is shown in Figure 8E, whereas an image cannot be formed in the forward direction due to the nonreflecting behavior of the structure (Figure 8D).
A couple of subsequent studies showed theoretically that unidirectional reflectionless light propagation can be realized in similar two-layer non-PT-symmetric lossy dielectric slab structures with modulation of the imaginary part of the refractive index of each slab , . Ge and Feng further pointed out that an optical reciprocity-induced symmetry, which is related to the amplitude ratio of the incident waves in the scattering eigenstates, can lead to an EP. Because optical reciprocity holds in general and does not rely on PT symmetry, the unidirectional reflectionlessness at EPs can be obtained in optical systems with unbalanced gain and loss and even in the absence of gain . In addition, Kang et al. designed a non-PT-symmetric ultrathin metamaterial to exhibit one-way zero reflection at EPs .
Forming an EP of the S matrix through tuning the geometric parameters of a structure rather than the refractive index profile was proposed in a non-PT-symmetric plasmonic waveguide-cavity system consisting of two metal-dielectric-metal (MDM) stub resonators side coupled to an MDM waveguide (Figure 9A) . Tuning the geometry rather than the refractive index can reduce the difficulty in the experimental realization of such structures. Among different plasmonic waveguide structures, MDM plasmonic waveguides are of particular interest , , , , , , , , , because they support modes with deep subwavelength scale over a very wide range of frequencies extending from DC to visible  and are relatively easy to fabricate , . The waveguide widths w, w1, and w2 are set to be 50, 20, and 100 nm, respectively (Figure 9A). The metal is silver and the dielectric is air. To obtain unidirectional reflectionless propagation, the MDM stub lengths L1, L2, as well as the distance between the stubs L, are optimized using the scattering matrix theory ,  to minimize the amplitude of the reflection coefficient in the forward direction |rf| at the optical communication wavelength of λ0=1.55 μm. Note that, if one tunes the refractive index of a material to form an EP, the optimized refractive index will be complex. However, if one tunes the geometric parameters to obtain an EP, these parameters are restricted to be purely real.
Figure 9B shows the reflection spectra for the structure of Figure 9A calculated for both forward and backward directions for L1=175 nm, L2=365 nm, and L=561 nm. The results verify that the optimized structure of Figure 9A is unidirectional reflectionless at f=193.4 THz (λ0=1.55 μm). Figure 9C and D shows the real and imaginary parts, respectively, of the eigenvalues of the scattering matrix S [Equation (1)] as a function of the distance L between the two MDM stub resonators. We observe that the real and imaginary parts of the two eigenvalues indeed collapse for L=561 nm. As discussed above, this is the optimal distance between the two stubs, which minimizes the reflection in the forward direction. In Figure 9C and D, we observe the level repulsion in the real parts of the eigenvalues, as well as the level crossing in their imaginary parts, which resembles a system also described by a non-Hermitian Hamiltonian matrix consisting of two coupled damped oscillators . In open quantum systems, a repulsion (crossing) for the real part of the energy and a crossing (repulsion) for the imaginary part of the energy in the 2D complex energy plane are required around EPs , . Unlike the Hermitian case, the levels approach each other in the form of a cusp rather than a smooth approach because of the plain square-root behavior of the singularity (Figure 9C) , .
Similar to other classical optical systems that have EPs , we observe that a generalized power decreasing phase and a generalized power increasing phase are divided by the EP at f=193.4 THz (Figure 9E). In addition, an abrupt phase change in the differential generalized power spectrum is observed at the EP as well (Figure 9E). These results are essentially due to the fact that the reflection coefficient in the forward direction rf approaches zero at the EP. Figure 9F shows that the phase of the reflection coefficient in the forward direction undergoes an abrupt π jump, when the frequency is crossing over the EP, which actually resembles the phase transition from the PT-symmetric phase to the PT broken phase in optical PT-symmetric systems , , , , , . Such an abrupt π-phase jump in the reflection coefficient in the forward direction confirms the existence of the EP in this plasmonic system and that the unidirectional reflectionlessness in the system is directly associated with this EP. In contrast, the phase of the reflection coefficient in the backward direction does not undergo an abrupt jump and varies smoothly with frequency. In addition, the reflection is not unidirectional if the system is lossless (Figure 9B). This is different from other classical optical analogues of quantum systems, such as the plasmonic analogue of electromagnetically induced transparency , , which can be realized in both lossless and lossy optical systems. In addition, the formation of an EP and the resulting unidirectional reflectionlessness can also be implemented using plasmonic waveguide-cavity systems based on other plasmonic two-conductor waveguides, such as 3D plasmonic coaxial waveguides , .
Unlike PT-symmetric systems, most non-PT-symmetric optical systems with modulation of the real or the imaginary part of the refractive index of the material typically exhibit unidirectional reflectionless propagation only within a very narrow wavelength range around the EP (see, for example, Figures 8C and 9B) , , , , . This is especially true for nonperiodic non-PT-symmetric systems, which are relatively easy to fabricate and more compact. Although Yang et al. demonstrated broadband unidirectional reflectionless light transport at an EP in a periodic ternary-layered structure consisting of lossy and lossless dielectrics , large-sized periodic structures are not easy to implement in densely integrated optical chips.
A compact non-PT-symmetric plasmonic waveguide-cavity system consisting of two MDM stub resonators with unbalanced gain and loss side coupled to an MDM waveguide was recently theoretically investigated (Figure 10A) . This non-PT-symmetric structure can exhibit unidirectional reflectionlessness in the forward direction at λ0=1.55 μm. Moreover, light reflection in the forward direction in this system is close to zero in a broad wavelength range.
where ω01 and ω02 are the resonance frequencies of the two resonators, 1/τi, i=1, 2, are the decay rates of the resonator mode amplitudes due to the power escape through the waveguide, 1/τ0i, i=1, 2, are the decay (growth) rates due to the internal loss (gain) in the resonators, and L is the distance between the two resonators. α and β are the real and imaginary parts, respectively, of the complex propagation constant of the fundamental mode in the waveguide at λ0=1.55 μm. The left and right stubs are filled with SiO2 doped with CdSe quantum dots (εA=4.0804−j0.6) and InGaAsP (εB=11.38+j0.41), respectively. The real parts of εA and εB are fixed, whereas the imaginary parts are tuned to satisfy Equation (8). In Figure 10B, we observe that not only the system exhibits unidirectional reflectionless light propagation at the resonance frequency of f0=193.4 THz (λ0=1.55 μm) but also the off-resonance reflection in the forward direction beyond f0=193.4 THz is significantly suppressed in a broad frequency range.
More interestingly, as shown in Figure 10B, the on-resonance reflection in the backward direction is unity. This is due to the fact that the right stub behaves as a lossless stub [ Equation (8)]; therefore, it acts as a perfect reflector at the resonant frequency (f0=193.4 THz). The gain of the material filling the right stub compensates the material loss in the metal. Thus, the on-resonance transmission for light incident from both the forward and backward directions is zero, and for light incident from the left, the on-resonance absorption is unity. We note here that the existence of an EP-induced unidirectional reflectionless mode with unity transmission in PT-symmetric systems with balanced gain and loss (Figure 2) is commonly referred to as unidirectional invisibility, whereas the existence of an EP-induced unidirectional reflectionless mode with zero transmission corresponds to unidirectional perfect absorption. Ramezani et al. also obtained unidirectional perfect absorption in passive Fano disk resonators at EPs based on a similar idea .
The unidirectional perfect absorption in the forward direction can be observed in the magnetic field distributions. When the waveguide mode is incident from the right (backward direction), there is no transmission, and the incident and reflected fields form a strong interference pattern (Figure 10D and F). On the contrary, when the waveguide mode is incident from the left (forward direction), there is hardly any reflection or transmission (Figure 10C and E). In addition, by cascading multiple non-PT-symmetric optical structures with different resonant absorption frequencies, and taking advantage of their broadband near-zero reflection property, an ultra-broadband near-total absorber can be realized. It should be noted that the proposed non-PT-symmetric waveguide-cavity system can also be realized using other nanophotonic structures such as microring and PhC cavities , . In such lossless structures, gain media are not required to implement the proposed waveguide-cavity systems, as there is no internal loss in the resonators.
Recently, Yang et al. performed the first direct measurement of asymmetric backscattering (reflection) in a microcavity. Their setup consists of an erbium-doped silica microtoroid whispering-gallery modes (WGM) resonator that allows for in- and out-coupling of light through two single-mode optical fiber waveguides (Figure 11A) . They have also previously reported PT-symmetry breaking in two coupled WGM silica microtoroid resonators with balanced gain and loss . Here, to probe the asymmetric backscattering of the WGMs, they used two silica nanotips as Rayleigh scatterers (Figure 11A). The microcavity is an open system, and the corresponding effective Hamiltonian is
which is, in general, non-Hermitian. The real parts of the diagonal elements Ωc are the frequencies, and their imaginary parts are the decay rates of the resonant traveling waves. The complex-valued off-diagonal elements A and B are the backscattering coefficients, which describe the scattering from the clockwise (cw) [counterclockwise (ccw)] to the counterclockwise (clockwise) propagating wave (Figure 11A). The complex eigenvalues of H are
and its eigenvectors are
When the first scatterer is introduced into the WGM volume, frequency splitting (Ω+ and Ω−) can be observed in the transmission spectra due to scatterer-induced modal coupling between the cw and ccw propagating modes. Subsequently, the relative position (i.e. relative phase angle β) and the size of the second scatterer are tuned by nanopositioners to bring the system to an EP (Figure 11B), which, as mentioned above, is a non-Hermitian degeneracy identified by the coalescence of the complex frequency eigenvalues (Ω+ and Ω−) and the corresponding eigenstates Ψ+=Ψ−. As a result, asymmetric backscattering (unidirectional reflectionlessness) of the WGMs can be achieved (A=0, or B=0).
In the absence of scatterers, a resonance peak appears in the transmission and no signal is observed in the reflection (backscattering), when the light is incident from the cw direction, as a result of wave vector matching (Figure 11C-i). Similarly, there is a resonance peak in the transmission and no signal in the reflection when the light is coming from the ccw direction (Figure 11D-i). In the presence of the first scatterer, two split resonance modes are observed in the transmission and reflection spectra regardless of whether the signal is coming from the cw or the ccw direction (Figure 11C-ii and D-ii). This implies that the field inside the resonator is composed of modes propagating in both cw and ccw directions due to the modal coupling between these two modes. Note that reflection in the cw and ccw directions is symmetric (red line in Figure 11C-ii and blue line in Figure 11D-ii). When the second scatterer is introduced and its position and size are tuned to form an EP, the transmission in the two different directions is still symmetric, whereas the reflection is asymmetric (red line in Figure 11C-iii and blue line in Figure 11D-iii). The reflection in the cw direction vanishes, whereas the reflection in the ccw direction supports a resonance peak. This unidirectional zero backscattering phenomenon can support chiral behavior , , , ,  and enable directional emission of a WGM microlaser at EPs. It is also noteworthy that the EP in this non-PT-symmetric microcavity system is induced by tuning of the geometric parameters, such as the size of scatterers and the relative phase angle between scatterers (Figure 11B), rather than tuning of the refractive index of the materials.
4 Conclusions and outlook
Asymmetric light transport is important for several key applications in photonic circuits , , , . In this paper, we reviewed unidirectional reflectionless light propagation at EPs. We first discussed the large body of recent works on unidirectional invisibility using PT-symmetric systems with balanced gain and loss. We then discussed how it is possible to achieve one-way zero reflection at EPs through a periodic passive PT-symmetric modulation of the dielectric permittivity. When additional material potentials, such as the magnetic permeability, are considered, additional degrees of freedom exist in establishing PT symmetry to obtain unidirectional reflectionlessness at EPs. In addition, EPs exist in a larger family of non-Hermitian Hamiltonians. As an example, a medium with a permittivity profile, which is an analytic function in the upper or lower half of the complex position plane, and its real and imaginary parts are related by the spatial Kramers-Kronig relations, shows unidirectional nonreflecting behavior. This finding is more general than previous results associated with PT symmetry. The proposed non-PT-symmetric optical systems also provide a simple way to realize asymmetric reflection by solely tuning the imaginary part of the permittivity of materials. In addition, systems that exhibit unidirectional perfect absorption have also been demonstrated. We finally reviewed unidirectional reflectionless light propagation at EPs in non-PT-symmetric structures through tuning of their geometric parameters rather than the refractive index profiles of the materials. These designs can reduce the difficulty in the experimental realization of such structures.
Unidirectional reflectionless light propagation is an intriguing phenomenon. Based on unidirectional invisibility in acoustics, a noninvasive, shadow-free, fully invisible acoustic sensor with PT symmetry was recently demonstrated . Unidirectional invisibility at EPs in optics may also have remarkable implications for noninvasive sensing. In addition, applying unidirectional perfect absorption designs in optical structures based on electro-optic, absorptive, and nonlinear materials with outstanding properties could advance the development of nanophotonic devices, such as switches, modulators, and devices for imaging. In addition, EPs in plasmons could lead to nanoscale photonic devices with novel functionalities.
The authors want to acknowledge financial support from the Foundation of the Key Laboratory of Optoelectronic Devices and Systems of the Ministry of Education and Guangdong Province (GD201601), the National Science Foundation (NSF) grants 1102301 and 1254934, the National Natural Science Foundation of China (NSFC) grants 61605252 and 61422506, and the Air Force Office of Scientific Research (AFOSR) grants FA9550-12-1-0471 and FA9550-17-1-0002.
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About the article
Published Online: 2017-05-20
Citation Information: Nanophotonics, Volume 6, Issue 5, Pages 977–996, ISSN (Online) 2192-8614, DOI: https://doi.org/10.1515/nanoph-2017-0019.
©2017, Georgios Veronis et al., published by De Gruyter, Berlin/Boston. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0