The ability to control the flow of light is of fundamental significance and desired for the next-generation of photonic integrated devices [1]. Inspired by band theory developed in solid state physics, its photonic analogies, such as photonic crystals [2], ring resonator arrays [3–5], and waveguide arrays [6–8], have sparked a wide range of applications by taking advantage of micro/nanofabrication technology. In particular, waves tunneling dynamics in coupled waveguides resembles that of electron hopping in crystals. Therefore, waveguide arrays are successfully utilized to simulate unique electron dynamics that may be difficult observed in electron crystal. Discrete diffraction [9], Bloch oscillation [10, 11], dynamic localization [12, 13], Anderson localization [14], and massless particle [15] were theoretically and experimentally demonstrated in different kinds of waveguides, which in turn provide new approaches to controlling light transport and localization. In addition, synthetical gauge fields can be generated by periodically bending waveguides [16], dynamical modulation of refractive index [17, 18], or using orbital coupling [6, 19], which give rise to Aharanov-Bohm caging [19], Fouquet topological insulator [20], high-order topological insulator, and semimetals [21, 22].

Recently, band theory in non-Hermitian photonic systems has also attracted a lot of attention with considering the influence of gain and loss. The early studies mainly focus on parity-time (PT) and anti-PT-symmetric systems where non-Hermiticity is introduced by on-site gain and loss [23, 24]. On the other hand, non-Hermiticity can also be produced by anisotropic coupling, which leads to intriguing non-Hermitian skin effect (NHSE) with a number of modes accumulated at system boundaries [25]. The discovery of NHSE deepens the previous understanding of bulk and topological edge modes in the sense that bulk modes can be localized but topological edge modes may become extended [26]. Therefore, the traditional bulk-boundary correspondence is broken down and non-Bloch band theory is subsequently developed to indicate the existence of topological modes. In addition, NHSE itself belongs to a topological phenomenon protected by spectral winding number, which develops a new way for manipulating light [27]. Abnormal phenomena were discovered with the aid of NHSE, such as light funneling [28], self-acceleration [29], self-healing [30], and anomalous single-mode lasing [31]. NHSE are reported in various platforms, such as quantum walks [32], cold atoms [33], electric circuits [34], acoustics [35], optical fibers [28], and photonic ring resonator arrays [4].

In this work, we present another platform by using modulated waveguide arrays to achieve NHSE. The waveguides undergo spatially refractive-index modulation and gain-loss modulation along light propagation direction at the same time, namely the complex modulation [36]. Such modulation yields one-way mode conversion in multimode waveguides, which was theoretically proposed in long-period grating and experimentally implemented in silicon waveguides [37–39]. Here, we utilize a set of modulated waveguides with each supporting two TE-polarized modes and further turn their refractive index to match the detuning of propagation constants of symmetric and anti-symmetric modes between adjacent waveguides. In this approach, we can create a non-Hermitian SSH photonic lattice that sustains NHSE with non-Hermitian anisotropic coupling induced by complex modulation. Since anisotropic coupling is unidirectional, NHSE inherits this direction-dependent behavior and tends to localize at different boundaries for different transport directions.

We start by investigating the anisotropic coupling aroused by complex modulation in a single photonic waveguide. Figure 1A shows the scheme of a waveguide where light propagates along z direction. The relative refractive indexes of cladding and core are denoted as n ₀ and n _core, respectively. For simplicity, we assume the cladding is air with n ₀ = 1. The waveguide holds two modes under TE polarization including symmetric and anti-symmetric modes, which propagation constants are denoted by k _zs and k _za. In Figure 1B, we plot the band structure of waveguides without modulation, namely, the propagation constant versus incident frequency. The insert illustrates the typical modal profiles of symmetric and anti-symmetric modes.

In general, the two modes are orthogonal to each other and thus the mode conversion is forbidden between them without modulation. To make them coupled, we add periodic modulation of complex permittivity along z direction. The modulation function reads as ∆ ε x , z = s i g n x − x 0 ∆ ε r cos 2 π z Λ + i ∆ ε i cos 2 π z Λ − ϕ (1) where x ₀ denotes center location of the waveguide, Δε _r and Δε _I are the amplitudes of the real and imaginary parts of the dielectric perturbation, Λ is the modulation period, and ϕ is the phase difference between real and imaginary modulation. The modulation profile is odd along x direction with respect to x ₀. When phase matching condition is satisfied with 2π/Λ = k _zs—k _za, the mode conversion is aroused between symmetric and antisymmetric modes. In the modulated region, the electric fields can be written as E x , z , t = a s z E s x exp i k z s − z ω t + a a z E a x exp i k z a − z ω t (2) where E _s,a denote the modal profiles of symmetric and anti-symmetric mode, a _s and a _a are their normalized amplitudes, respectively. When light is injected from the left side of the waveguide (forward direction), the real modulation is ahead of the imaginary modulation, and the phase difference is positive ϕ. By utilizing slowing varying approximation, one can derive the coupled mode equation in the modulated region for the forward propagation as follows − i d d a s a a = 0 c r + i c i exp i ϕ c r + i c i exp − i ϕ 0 a s a a (3) where the coupling coefficients are c r , i = 1 2 ε 0 ω ∫ ∆ ε r , i E s * x E a x d x (4)

The result indicates the complex modulation leads to complex coupling between symmetric and anti-symmetric modes. The phase difference ϕ between real and imaginary modulation acts as an effective gauge potential [40]. Its interaction with complex coupling finally leads to anisotropic coupling. It becomes more straightforward with considering a special case as ϕ = π/2, corresponding to PT-symmetric modulation [41]. In this case, the coupled mode equation reduces to − i d d a s a a = 0 c r + c i c r − c i 0 a s a a (5) where the off-diagonal elements are different. When the real and imaginary modulations have the same strength with Δε_r = Δε_i, the two coupling coefficients are equal with c _r = c _i. In this case, it only allows mode transition from antisymmetric to symmetric modes while the inverse conversion is forbidden.

In contrast, for the backward propagation, the imaginary modulation is ahead of the real modulation. Then the effective gauge potential becomes negative with ϕ = −π/2. As a result, the coupled mode equation for the right-side injection reads as − i d d a s a a = 0 c r − c i c r + c i 0 a s a a (6) with off-diagonal elements switched. As c _r = c _i, the mode conversion is reversed. The power can couple from symmetric to antisymmetric mode. However, no energy transfers from anti-symmetric mode to symmetric one. The one-way coupling behaviour is schematically indicated by the arrows in Figure 1B.

We further place an array of modulated multimode waveguides to realize a non-Hermitian SSH lattice, as shown in Figure 1C. Figure 1D plots the propagation constants of two modes as a function of the refractive index n _core with incident wavelength λ = 1.55 μm. The width of each waveguide is fixed at w = 1 μm. The blue and red lines stand for symmetric and antisymmetric modes, respectively. The two modes supported in a single waveguide are coupled by the complex modulation. We further adjust the refractive index of every waveguide such that the symmetric and anti-symmetric modes between adjacent waveguides have the same propagation constants and can be coupled. We first set the refractive index of the first waveguide to n _core = 1.5, then calculate the effective refractive index of the two modes in this waveguide, which are n _eff = 1.403 for symmetric mode and 1.116 for antisymmetric mode, respectively. In order to make the second waveguide couple with the previous waveguide, we should adjust the refractive index of second waveguide such that the antisymmetric mode of second waveguide is the same as that of the symmetric mode of the first waveguide. Then, the refractive index of the second waveguide is determined to be n _core = 2.084. Following this way, we can determine all refractive index of every waveguide. The refractive index from upper to lower waveguides is set to n _core = 1.50, 1.80, 2.08, 2.35, 2.598, 2.831, 3.053, 3.265, and 3.467. The materials for the parameters adopted can be SiO₂ and Si. The refractive indexes of Si and SiO₂ are 1.45 and 3.45, which cover the range of refractive index used above. Therefore, we could use the hybrid waveguide made by Si and SiO₂ to design the proposed model. In this way, we could realize an effective coupled photonic lattice with non-Hermitian coupling, as indicated by the red and blue dots in Figure 1D. We have briefly shown the periodically modulation results in a coupled mode equation with complex coupling for mode conversion process. More intuitively, we indeed integrate on-site gain and loss into the waveguide. As the on-site dissipation is periodically varied, it can provide a wavevector to meet the mismatch between symmetric and antisymmetric modes. In this way, the two modes can be coupled and the effective coupling coefficients are imaginary. This complex coupling occurs in the momentum space between symmetric and anti-symmetric modes. We emphasis that the real-space coupling, that is, the couplings between neighboring waveguides, are still real-valued. To further consider the unidirectional coupling generated by complex modulation, the proposed SSH lattice is direction-dependent since it inherits the unidirectional behaviour. The effective lattices for forward and backward directions are shown in Figures 1E,F, respectively. The intracell couplings are inversed for opposite directions.

To further gain insights into the unidirectional coupling, we perform full wave simulation based on finite element method. As a comparison, we present the transport of light as only the real part of dielectric permittivity is modulated. The distributions of electric field (E _y ) are shown in Figure 2A where light with wavelength λ = 1.55 μm is launched from the left side of waveguide, corresponding to forward propagation. Other parameters are set as w = 1 μm, n _core = 1.5, and n ₀ = 1. The propagation constants for symmetric and antisymmetric modes are figured out to be k _zs = 5.69 μm⁻¹ and k _za = 4.52 μm⁻¹, respectively. The modulation period is Λ = 2π/(k _zs − k _za) to match the detuning of two modes and the modulation amplitude is Δε _r = 0.1. The insert plots the effective lattice in which the transition is bidirectional. The results show that the antisymmetric mode at the incidence evolves into symmetric mode and then transfers back to antisymmetric mode. The coupling is bidirectional and can be also reflected by plotting mode amplitudes, as indicated by the blue line for symmetric and red line for antisymmetric mode in Figure 2B. At propagation distance z = 25 μm, we see a _a = 0 and a _s = 1, implying all light completely converts into symmetric modes. As propagation distance increases, a _a increases and a _s decreases, indicating light transfers back to antisymmetric modes. The same behaviour applies in backward injection due to the reciprocity in momentum space [42]. Figure 2C shows the field distributions in forward propagation as the waveguide is under complex modulation. The real and imaginary modulation strength is equal as Δε _I = Δε _r and the phase difference is ϕ = π/2. The coupling is completely unidirectional, as shown in the inset of Figure 2C. The injection of antisymmetric mode will stimulate symmetric mode. However, it will not transfer back. As a result, the symmetric mode is dominated at the output. We also plot the amplitudes of two modes as a function of propagation distance in Figure 2D. The amplitude of symmetric mode linearly increases (blue line) while the antisymmetric mode (red line) stays unchanged. For backward propagation, as shown in Figure 2E, the injected antisymmetric mode remains unchanged and cannot transfer to symmetric mode. This can be seen from mode amplitudes as well, as plotted in Figure 2F. The amplitude for antisymmetric mode (red line) keeps at unity for different z while it is vanished (blue line) for symmetric mode. Such one-way mode conversion clearly reflects the direction-dependent coupling due to complex modulation.

We emphasis that we only use spatial modulation without time modulation. As a result, the proposed model does not break Lorentz reciprocity and cannot be used for optical isolators. We surely obtain asymmetric behavior in mode conversion. However, it is not necessary to break Lorentz reciprocity as our system still respects t _ij = t _ji with t _ij denoting transmission coefficients from j mode into i mode. By involving time modulation, Lorentz reciprocity can be broken and the transmission coefficients fulfill t _ij ≠ t _ji. In this case, it is possible to construct an isolator.

As discussed above, a non-Hermitian version of SSH model can be realized by couple a set of modulated waveguides. We now analyse the eigenvalues and eigenvectors to clearly show the appearance of NHSE in the proposed systems. We consider the effective gauge potential is ϕ = π/2. Then the Bloch Hamiltonian for forward direction corresponding to Figure 1E is given as follows H f k = 0 c r + c i + c 2 e i k c r − c i + c 2 e − i k 0 (7) with c _r ± c _i denoting intracell coupling, c ₂ signifying intercell coupling, and k representing Bloch momentum. Such a non-Hermitian model has been studied in various systems, exhibiting NHSE under OBC and non-Bloch bulk-boundary correspondence [26, 43, 44]. Here we focus on NHSE, which can be indicated by eigenvalue spectra. Figures 3A–C plot the eigenvalue spectra on the complex plane for three different anisotropic couplings with c _i = 0, c _r/2, and c _r, respectively. Other parameters are fixed at c _r = 1 and c ₂ = 0.5. The blue dots and red circles stand for PBC and OBC, respectively. As c _i = 0, the intracell coupling is isotropic and the system is Hermitian. The spectra for OBC and PBC are real and almost identical to each other, as shown in Figure 3A. In this case, there is no NHSE under OBC. The corresponding distributions of eigenmodes are plotted in Figure 3D. The modes are extended throughout all sites. As c _i = c _r/2, the periodic spectrum forms closed loops in the complex plane. In contrast, the spectrum for OBC collapses into lines that resides within the loops for PBC. This is a typical feature of NHSE. According to bulk-boundary correspondence based on spectral winding, NHSE will take place under OBC if periodic spectra enclose any area. The spectral winding is given by W = ∫ − π π d k 2 π ∂ k arg H f k − E b (8) where E _b is any base point. The spectral winding number is to characterize the number of times of complex eigenvalues encircling E _b. The non-zero winding number indicates NHSE will appear in the system under OBC. If periodic spectra enclose any non-zero area, the base point can be chosen within the loop, which leads to W ≠ 0. In Figure 3E, we plot the distributions of eigenmodes, which are accumulated at the left boundary and exponentially decrease with distance away from the boundary. As we further increase the anisotropic degree with c _i = c _r, the enclosed area of periodic spectrum increases, as shown Figure 3C. NHSE effect becomes stronger in the sense that eigenmodes experience stronger localization at the left boundary, as shown in Figure 3F.

For backward direction, the effective gauge field switches to negative value and the intracell coupling changes as well. The Bloch Hamiltonian corresponding to Figure 1F is given by H b k = 0 c r − c i + c 2 e i k c r + c i + c 2 e − i k 0 (9)

The spectra for backward direction are similar to that of forward propagation. However, the location of skin modes is reversed. Figures 3G–I plot the distributions of eigenmodes for backward direction for c _i = 0, c _r/2, and c _r, respectively. As c _i = 0, as shown in Figure 3G, there are no skin modes. The eigenmodes are distributed throughout all sites, just like forward case. In contrast, when c _i = c _r/2 and c _i = c _r, as shown in Figures 3G,H, one can observe NHSE and skin modes are exponentially accumulated at the right boundary.

The NHSE and the direction-dependent behavior can be directedly viewed from light propagation by full wave simulation. In Figure 4, we employ nine modulated waveguides to achieve a SSH lattice. Each waveguide underlies complex modulation with ϕ = π/2. The refractive index of each waveguide is the same as that in Figure 1D. To generate the same coupling strength between symmetric and antisymmetric modes in the same waveguide, the modulation amplitudes of different waveguides should be adjusted. From upper to lower, the modulation amplitudes of dielectric permittivity are Δε _r = 0.12, 0.13, 0.15, 0.16, 0.18, 0.20.0.21, 0.23, and 0.24 with the modulation vectors 2π/Λ = 0.287k ₀, 0.303k ₀, 0.288k ₀, 0.270k ₀, 0.253k ₀, 0.238k ₀, 0.226k ₀, 0.215k ₀, and 0.204k ₀. As a result, the intracell couplings are identical, corresponding to c _r = 0.0628 μm⁻¹. On the other hand, the spatial spacing should be altered as well to make intercell couplings identical. The spacing from upper to bottom is d = 1.59 μm, 1.41 μm, 1.30 μm, 1.24 μm, 1.20 μm, 1.17 μm, 1.14 μm, and 1.12 μm, which leads to coupling coefficient c ₂ = 0.044 μm⁻¹. In this way, a photonic SSH lattice is created. Then, we launch light from the second waveguide at the bottom with antisymmetric modes. Figures 4A–C show the light propagation for forward propagation as Δε _i = 0, Δε _r/2, and Δε _r, respectively. As Δε _i = 0, light spreads and more and more waveguides are excited during the propagation, displaying discrete diffraction [Figure 4A]. As Δε _i = Δε _r/2, light tends to localize at the lower boundary with the aid of NHSE [Figure 4B]. As we further increase imaginary modulation with Δε _i = Δε _r, the localization becomes stronger [Figure 4C]. These results indicate the skin modes tend to localize light at the bottom of waveguide array. We further inject light from the right side, corresponding to backward propagation, as shown in Figures 4D–F. As Δε _i = 0 [Figure 1D], there is no NHSE and light spreads during the evolution, just the same as Figure 1A. In contrast, in the presence of NHSE, light tends to localize at the upper boundary for backward propagation [Figures 4E,F], significantly different from the forward direction. The results clearly show the direction-dependent behavior of NHSE using complex modulation. The wave propagation in Figure 4F is not very well confined into a single waveguide for the backward propagation. It may be caused by the following two reasons. Firstly, the input ports locate at the lower boundary, which is just near the location boundary for forward propagation but far from location boundary for backward injection. Secondly, the designed waveguide array is not perfectly equivalent to a tight-binding model and the coupling coefficients for the backward direction are not completely vanished. Anyway, the results can still reflect the unidirectional localization of NHSE.

In conclusion, we have proposed NHSE in waveguide arrays based on complex modulation. The complex modulation could arouse non-Hermitian coupling and effective gauge fields between symmetric and anti-symmetric modes, which interaction finally leads to anisotropic coupling. By coupling a set of waveguides and making the propagation constants of neighboring waveguides matched, we successfully create a photonic SSH lattice sustaining NHSE. Since the effective gauge fields switch its sign for opposite propagation directions, the anisotropic coupling changes its relative strength as well. Therefore, the localization direction of NHSE is direction-dependent in the sense that skin modes are accumulated to locate at the lower boundary for forward direction and upper boundary for backward direction. The proposed model may be experimentally implemented in dielectric waveguides. The modulation of real part of refractive index can periodically embed sinusoidal-shaped structure at the top of the silicon waveguide, while the imaginary part of modulation is realized by integrating periodically curved Ge or Cr layers [38]. In order to match the refractive index of adjacent waveguides, one can adjust the width and the height of Si. The difficulty may lie in how to fabricate a large-scale waveguide array together with each waveguide under complex modulation with different parameters. As NHSE is a topological phenomenon, the accuracy requirements of some parameters can be relaxed such as coupling coefficients, waveguide spacing, and the width of waveguides. We believe the proposed model can be realized soon. Here we only discussed skin modes but topological modes can also be explored in these systems to detect non-Bloch bulk-boundary correspondence and extended isolated modes [45]. Furthermore, the spatial modulation does not break optical reciprocity. The similar mechanism can be extended to dynamically modulated waveguides with involving temporal modulation. One could construct a special photon transition where transition can occur between two modes at different frequency for forward direction. However, for backward propagation, mode transition is forbidden due the wavevector mismatch, which may give rise to non-reciprocal transport of skin modes. Our results pave a way to chip-scale localization with the aid of NHSE and may find potential applications in non-magnetic unidirectional devices.