Broadband unidirectional surface plasmon polaritons with low loss

Jinhua Yan; Qian Shen; Hang Zhang; Senpeng Li; Haiwei Tang; Linfang Shen

doi:10.1364/OE.504997

1. Introduction

Topological electromagnetics [1–23] has drawn much attention in both fundamental and applied researches. One of the most intriguing aspects in this research field is the possibility of realizing truly unidirectional wave propagation channels, which represent a highly nontrivial regime of wave propagation and may enable extreme and counterintuitive optical effects. The guiding modes in such channels are allowed to propagate only in one direction, and they can be immune to backscattering against defects because of the absence of backward-propagating mode in the system. Such unidirectional electromagnetic (EM) modes (or channels) were first proposed by Raghu and Haldan [1] as analogs of quantum Hall edge states [24] in photonic crystals (PhCs) [25]. It was predicted that such modes can be sustained by the edges of two-dimensional (2D) PhCs with topologically nontrivial bandgaps, which are made of magnetic-optical (MO) materials, and where the time-reversal symmetry is broken by applying a dc magnetic field. The existence of such modes was later verified both theoretically and experimentally by Wang et al. using MO PhCs in the microwave regime [3,4]. Since then, different approaches for achieving unidirectional wave propagation, including directional excitations, have been proposed [5,11–15], and among them, the one [5] based on nonreciprocal surface plasmon polaritons (SPPs) [26,27] seems to be most attractive owing to its robustness in mechanism and simpleness in configuration.

Unidirectional SPPs (USPPs) may exist in plasmonic platforms biased by a dc magnetic field, which breaks time-reversal symmetry of the system [5–8]. The propagation characteristic of USPPs closely depends on the material configuration and frequency [8]. Up to now, two types of USPPs have been reported. The first type of USPP, known as conventional surface magnetoplasmon (referred to as type-I USPP) [27], is supported by an interface between a magnetized plasmonic material and transparent dielectric [5–7]. USPP of this type occurs in the lower bandgap of bulk-modes in the magnetized plasmonic material and its unidirectional window results from the asymmetry between upper frequency cutoffs for the positive and negative dispersion branches. The second type of USPP (referred to as type-II USPP) is supported by an interface between a magnetized plasmonic material and an opaque isotropic medium [8]. Type-II USPP occurs in the upper bulk-mode bandgap of the magnetized plasmonic material, and its unidirectional window is determined by different lower frequency cutoffs of the positive and negative dispersion branches. However, recent studies have demonstrated that when considering the nonlocal effect of plasmonic material [28], type-I USPP loses its strict unidirectionality, because the dispersion curves no longer exhibit a flat asymptotic behavior (at large wavenumbers) and their upper-frequency cutoffs vanish [9]. But type-II USPP still preserves its unidirectional nature in the presence of nonlocality due to the nontrivial topological property of the upper bulk mode bandgap [8,10], which is characterized by a nonzero gap Chern number [29]. Nevertheless, type-II USPP becomes a leaky mode, since the nonlocality induces an additional bulk mode in the plasmonic material and thus makes the upper bandgap being incomplete.

It is now clear that true USPP really exists, but is accompanied by the issue of leakage caused by nonlocality. For the example in [8], it was observed that USPP at terahertz frequencies has a propagation length of smaller than two wavelengths only due to its leakage loss. Obviously, the leakage of type-II USPP should not be neglected, as it would hinder its practical applications. It would be naturally desired to eliminate this leakage by truncating the plasmonic material using a bandgap material, such as metals for the terahertz regime or photonic crystals (PhCs) for the visible regime. However, in this scenario, based on the mechanism of total internal reflection (TIR), the plasmonic material layer would support bidirectional-propagating modes, which are discretized additional bulk modes with different orders in transverse resonance. These TIR modes generally have lower frequency cutoffs determined by the thickness of the plasmonic-material layer. But our analysis shows that for the example mentioned above, when the bidirectional modes are all suppressed within the upper bulk-mode bandgap, the plasmonic material (i.e., semiconductor) layer becomes so thin that it cannot sustain USPP anymore, i.e., the USPP mode vanishes as well. Here, we should indicate that the spatial dispersion effect of the magnetized semiconductors is somewhat similar to that of wire metamaterials: it induces an additional bulk mode in the material [30–32]. For wire metamaterials, some strategies that effectively suppress the spatial dispersion have been proposed [30,31]. However, it is very difficult to apply these strategies to gyroelectric semiconductors, because the nonlocal effects for the two classes of materials are physically different. The nonlocal effect is the coupling between neighbouring wires (i.e., transmission-line effect) for wire metamaterials, while it is the pressure gradient (acting on free carriers) for gyroelectric semiconductors. So it is necessary to develop different classes of USPPs that have no leakage or very low leakage loss.

In this paper, based on nonreciprocal plasmonic platform, we develop a different type of USPP that is robust to nonlocality and simultaneously exhibits a very small leakage. In the terahertz regime, USPP of this class can be supported by a semiconductor-dielectric-metal layered structure, where the dielectric layer is a heterostructure of silicon (Si) and air. Compared to type-II USPP, the nonlocality-induced leakage of our proposed USPP is substantially reduced since the fraction of the modal energy distributed in the cladding layer of the nonreciprocal plasmonic platform is largely increased. Moreover, except for the leakage loss, its absorption loss due to the semiconductor dissipation is also significantly decreased. Besides, completely different from type-I USPP supported by a semiconductor-dielectric interface, which exists for larger wavenumbers in the lower bulk-mode bandgap, the proposed USPP exists for small wavenumbers in the upper bulk-mode bandgap with nontrivial topological property, hence the nonlocal effect does not affect its dispersion feature at all.

2. Characteristic of USPP in the local model

The plasmonic waveguide we consider is schematically illustrated in Fig. 1(a), which is a semiconductor-silicon-air-metal layered structure. In this waveguide, the cladding on the semiconductor is a dielectric heterostructure composed of silicon ($\varepsilon _{r1}=11.68$) and air ($\varepsilon _{r2}=1$), and its thickness is given by $d=d_1+d_2$, where $d_1$ and $d_2$ are the thicknesses of the silicon and air layers, respectively. In the terahertz regime, the metal layer can be approximated as perfect electric conductor (PEC). In order to break time-reversal symmetry of the waveguide system, an external magnetic field is applied in the $z$ direction, thus the semiconductor in it exhibits gyroelectric anisotropy, with a relative permittivity tensor in the form [27],

(1)$$\bar{\bar{\varepsilon}}_s=\left[ \begin{array}{ccc} \varepsilon_1 & i\varepsilon_2 & 0\\ -i\varepsilon_2 & \varepsilon_1 & 0\\ 0 & 0 & \varepsilon_3 \end{array}\right],$$

with

\begin{aligned}\varepsilon_1&=\varepsilon_{\infty}\left(1-\frac{\omega_p^2}{\omega^2-\omega_c^2}\right),\\ \varepsilon_2 &= \varepsilon_{\infty}\frac{\omega_c\omega_p^2}{\omega(\omega^2-\omega_c^2)},\\ \varepsilon_3 &=\varepsilon_{\infty}\left(1-\frac{\omega_p^2}{\omega^2}\right),\end{aligned}

where $\omega _p$ is the plasma frequency, $\omega _c$ is the electron cyclotron frequency, and $\varepsilon _{\infty }$ is the high-frequency relative permittivity. In this paper, the semiconductor is assumed to be InSb with $\varepsilon _{\infty }=15.6$ and $f_p=2$ THz ($f_p=\omega _p/2\pi$) [7], and the parameter $\omega _c$ related to the external magnetic field is set at $0.25\omega _p$, i.e., $\omega _c=0.25\omega _p$. In the magnetized semiconductor, the bulk modes have a dispersion relation of $k=\sqrt {\varepsilon _{\mathrm {v}}}k_0$ for transverse-magnetic (TM) polarization, where $k$ is the propagation constant, $k_0$ is the vacuum wavenumber, and $\varepsilon _{\mathrm {v}}=\varepsilon _1-\varepsilon _2/\varepsilon _1^2$ being the Voigt permittivity. Thus, the magnetized semiconductor has two bandgaps with $\varepsilon _{\mathrm {v}}<0$. The lower bulk-mode bandgap is below a certain frequency $\omega _a$, which is given by $\omega _a=\sqrt {\omega _c^2/4+\omega _p^2}-\omega _c/2$, and it exists even in the absence of external magnetic field. The upper bulk-mode bandgap is between $\omega _r$ and $\omega _b$, which are given by $\omega _r=\sqrt {\omega _c^2+\omega _p^2}$ and $\omega _b=\sqrt {\omega _c^2/4+\omega _p^2}+\omega _c/2$, and it vanishes if external magnetic field is absent. For $\omega _c=0.25\omega _p$, this bandgap is centered at $1.08\omega _p$ and has a bandwidth of nearly $0.1\omega _p$. The upper bulk-mode bandgap opened by the magnetic field is of our interest, as it exhibits nontrivial topological properties, characterized by a nonzero gap Chern number [10,29]. It is desired that based on topologically nontrivial bandgap, USPP in the waveguide is topologically protected and thus robust to the nonlocal effects.

Fig. 1. (a) Schematic of the proposed USPP waveguide, which is a semiconductor-silicon-air-metal layered structure under an external magnetic field. (b) Typical dispersion diagram for the proposed waveguide, where $d_1=\lambda _p/75$ and $d_2=0.1\lambda _p$. (c) Schematic of the previous USPP waveguide, which is a magnetized semiconductor-opaque isotropic material layered structure. (d) Typical dispersion diagram for the previous waveguide. The solid lines in (b) and (d) are the SPP dispersion curves, and the shaded areas represent the zones of bulk mode in the magnetized semiconductor. The dashed lines in (b) represent light lines in air. The basic parameters of the magnetized semiconductor are $\varepsilon _{\infty }=15.6$ and $\omega _c=0.25\omega _p$.

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The field of SPP in the considered waveguide is TM polarized. Thus, in the semiconductor ($y<0$), the magnetic field of SPP has a nonzero component

(2)$$H_z=A\exp(\gamma y)\exp(ikx),$$

where $k$ is the propagation constant, and $\gamma =\sqrt {k^2-\varepsilon _{\mathrm {v}}k_0^2}$. In the dielectric layers (silicon and air), the magnetic field component $H_z$ is expressed as

(3)$$\begin{aligned}H_z=[B_1\exp(-\alpha_1 y)+B_2\exp(\alpha_1 y)]\exp(ikx)\quad &{\rm{for}}\quad 0\le y\le d_1,\\ H_z=[C_1\exp(-\alpha_2 y)+C_2\exp(\alpha_2 y)]\exp(ikx)\quad &{\rm{for}}\quad d_1< y\le d, \end{aligned}$$

where $\alpha _i=\sqrt {k^2-\varepsilon _{r_i}k_0^2}~~(i=1,2)$. Using Maxwell’s equations, the nonzero components ( $E_x$ and $E_y$) of the electric field in each layer can be directly derived from $H_z$. According to the boundary conditions of electric and magnetic fields, which require the continuity of $E_x$ and $H_z$ at $y=0$ and $d_1$, and $E_x=0$ at $y=d$, the dispersion relation of SPP can be derived as

(4)$$\gamma-\frac{\varepsilon_2}{\varepsilon_1} k +\varepsilon_v \frac{\alpha_1 \tanh \left(\alpha_1 d_1\right) / \varepsilon_{r_1}+\alpha_2 \tanh \left(\alpha_2 d_2\right) / \varepsilon_{r_2}}{1+\left(\varepsilon_{r_1} \alpha_2/\varepsilon_{r_2} \alpha_1\right) \tanh \left(\alpha_1 d_1\right) \tanh \left(\alpha_2 d_2\right) }=0.$$

In the special cases of $d_1=0$ or $d_2=0$, the above dispersion equation is simplified to

(5)$$\gamma-\frac{\varepsilon_2}{\varepsilon_1} k+\frac{\varepsilon_v}{\varepsilon_r}\alpha\tanh (\alpha d)=0,$$

where $\varepsilon _r=\varepsilon _{r2}$ and $\alpha =\alpha _2$ for $d_1=0$, or $\varepsilon _r=\varepsilon _{r1}$ and $\alpha =\alpha _1$ for $d_2=0$, and this is just the dispersion relation for conventional surface magnetoplasmons in the magnetized semiconductor-dielectric-metal layered structure [7].

Figure 1(b) shows the dispersion diagram of SPPs for the considered waveguide with $d_1=2$ $\mu m$ and $d_2=15$ $\mu m$. In the lower bulk-mode bandgap of the magnetized semiconductor, both forward and backward SPPs exist, and they have different asymptotic frequencies, at which $k\to \pm \infty$. As the asymptotic frequency for the negative dispersion branch ($k<0$) is larger than the upper-frequency cutoff for the positive dispersion branch ($k>0$), there exists a frequency window where SPP is only allowed to propagate backward. However, flat asymptotic dispersion behavior is nonphysical since it violates thermodynamics, as demonstrated in [9]. When the nonlocal effects are considered, such flat asymptotic dispersion will vanish at large wavenumbers, then upper-frequency cutoff no longer occurs for both forward and backward SPPs. Hence, unidirectional propagation of SPP in the lower bulk-mode bandgap is not physically strict. In the upper bulk-mode bandgap of the magnetized semiconductor, there only exists a forward-propagating SPP with small $k$ values. Evidently, this USPP mode is robust to the spatial dispersion caused by nonlocality. In what follows, we will focus on USPP in the upper bulk-mode bandgap. For comparison, Fig. 1(d) presents the dispersion diagram for the previously reported structure as illustrated in Fig. 1(c). Though USPPs in the two structures have similar dispersion properties, their other modal properties are quite different especially when the nonlocal effects are considered, as we will see below.

In the upper bulk-mode bandgap of the magnetized semiconductor, our considered structure may also support guiding modes based on the TIR mechanism. Generally, these TIR modes are bidirectionally propagating, so they should be suppressed in the system. The dispersion relation for TIR modes can be obtained from Eq. (4) by substituting $\alpha _1=ip_1$ and $\alpha _2=ip_2$, and then it becomes

(6)$$\gamma-\frac{\varepsilon_2}{\varepsilon_1} k -\varepsilon_v \frac{p_1 \tanh \left(p_1 d_1\right) / \varepsilon_{r_1}+p_2 \tanh \left(p_2 d_2\right) / \varepsilon_{r_2}}{1-\left(\varepsilon_{r_1} p_2/\varepsilon_{r_2} p_1\right) \tanh \left(p_1 d_1\right) \tanh \left(p_2 d_2\right) }=0,$$

where $p_1=\sqrt {\varepsilon _{r_1} k_0^2-k^2}$ and $p_2=\sqrt {\varepsilon _{r_2} k_0^2-k^2}$. As each TIR mode corresponds to a transverse resonance of certain order in the dielectric heterostructure layer, it generally has a lower frequency cutoff determined by the parameter $d$. Though the TIR modes in the present guiding system are nonreciprocal, their cutoff points should be very close to $k=0$, so their existence conditions can be derived by analyzing the dispersion equation at $k=0$. At $k=0$, $\gamma =\sqrt {-\varepsilon _v}k_0$, $p_1=\sqrt {\varepsilon _{r_1}}k_0$, $p_2=\sqrt {\varepsilon _{r_2}}k_0$, and the dispersion equation reduces to

(7)$$1+\sqrt{-\frac{\varepsilon_v}{\varepsilon_{r_1}}} \frac{\tan \left(\sqrt{\varepsilon_{r_1}} k_0 d_1\right)+\sqrt{\varepsilon_{r_1} / \varepsilon_{r_2}} \tan \left(\sqrt{\varepsilon_{r_2}} k_0 d_2\right)}{1-\sqrt{\varepsilon_{r_1} / \varepsilon_{r_2}} \tan \left(\sqrt{\varepsilon_{r_1}} k_0 d_1\right) \tan \left(\sqrt{\varepsilon_{r_2}} k_0 d_2\right)}=0.$$

To suppress TIR modes, the following condition is required

(8)$$\frac{\tan \left(\sqrt{\varepsilon_{r_1}} k_0 d_1\right)+\sqrt{\varepsilon_{r_1} / \varepsilon_{r_2}} \tan \left(\sqrt{\varepsilon_{r_2}} k_0 d_2\right)}{1-\sqrt{\varepsilon_{r_1} / \varepsilon_{r_2}} \tan \left(\sqrt{\varepsilon_{r_ 1}} k_0 d_1\right) \tan \left(\sqrt{\varepsilon_{r_2}} k_0 d_2\right)}\ge 0.$$

Let's first consider the special cases of $d_1=0$ or $d_2=0$. In these cases, the above equation is simplified to $\tan (\sqrt {\varepsilon _r}k_0 d)\ge 0$, which requires $\sqrt {\varepsilon _r}k_0 d \le \pi /2$, here $\varepsilon _r=\varepsilon _{r_1}$ and $d=d_1$, or $\varepsilon _r=\varepsilon _{r_2}$ and $d=d_2$. Therefore, for the general case of $d_1\ne 0$ and $d_2\ne 0$, it is reasonable to assume that $\sqrt {\varepsilon _{r_1}}k_0 d_1 \le \pi /2$ and $\sqrt {\varepsilon _{r_2}}k_0 d_2 \le \pi /2$, then the suppression condition for the TIR modes can be written as

(9)$$\tan\left(\sqrt{\varepsilon_{r_2}} k_0 d_2\right)\le \sqrt{\frac{\varepsilon_{r_2}}{\varepsilon_{r 1}}} \cot \left(\sqrt{\varepsilon_{r_1}} k_0 d_1\right).$$

Let’s take $k_0=\omega _b/c$, where $\omega _b$ is the upper limit of the semiconductor bandgap of our interest, then the critical thickness $d_{2c}$ of the dielectric layer $\varepsilon _{r_2}$ is obtained as

(10)$$d_{2 c}=\frac{1}{\sqrt{\varepsilon_{r_2}}} \frac{c}{\omega_b} \tan ^{{-}1}\left[ \sqrt{\frac{\varepsilon_{r_2}}{\varepsilon_{r_1}}} \cot \left(\sqrt{\varepsilon_{r_1}} \frac{\omega_b}{c} d_1\right)\right].$$

Evidently, the critical thickness for the dielectric heterostructure layer is $d_c=d_{2c}+d_1$, where $d_1\le \pi c/2\sqrt {\varepsilon _{r_1}}\omega _b$, and it is a function of $d_1$, as illustrated in Fig. 2(a). $d_c$ decreases with $d_1$, and it has a maximum of $d_c=\pi c/(2\omega _b)$ at $d_1=0$, which corresponds to the special case of all air between the semiconductor and metal. $d_c$ decreases to a minimum of $d_c=\pi c/(2\sqrt {\varepsilon _{r 1}}\omega _b)$ in another special case of $d_2=0$, in which it is all silicon between the semiconductor and metal. How to choose the value of $d_1$ for the present structure? The goal of our design is to distribute the energy of USPP to the dielectric layers as much as possible, thus the loss of USPP falls to its lowest level. Note that the semiconductor as a dispersive medium is inherently lossy. So, in the local model, USPP still suffers absorption loss caused by the semiconductor dissipation. The optimal value of $d_1$ can be determined by analyzing the attenuation coefficient of USPP. For a lossy magnetized semiconductor, the tensor elements $\varepsilon _1$ and $\varepsilon _2$ in Eq. (1) become [33]

\begin{aligned}\varepsilon_1&=\varepsilon_{\infty}\left\{1-\frac{(\omega+i\nu)\omega_p^2}{\omega[(\omega+i\nu)^2-\omega_c^2]}\right\},\\ \varepsilon_2&=\varepsilon_{\infty}\frac{\omega_c\omega_p^2}{\omega[(\omega+i\nu)^2-\omega_c^2]},\end{aligned}

where $\nu$ is the electron scattering frequency. In the lossy case, the propagation constant of USPP becomes a complex, i.e., $k=k_r+ik_i$, where $k_r$ and $k_i$ are real-valued. We calculate attenuation coefficient ($k_i$) of USPP for various $d_1$ values. In the calculations, the frequency is set at $f=1.08f_p$ and the loss parameter at $\nu =0.002\omega _p$, and we take $d_2=d_{2c}$ for each $d_1$ value according to Eq. (10). The calculated results are plotted in Fig. 2(b). The variation of $k_i$ with $d_1$ is well consistent with that of the fraction ($\eta _s$) of the modal energy distributed in the semiconductor, as illustrated in Fig. 2(c). This implies that the loss of USPP can be controlled by distributing modal energy between the semiconductor and dielectric layers. Note that for the magnetized semiconductor, the EM energy density is given by $U_{em}=(\varepsilon _0/4){\bf E^*}\cdot d(\omega {\bar {\bar {\varepsilon }}})/d\omega \cdot {\bf E}+(\mu _0/4)|{\bf H}|^2$. From Fig. 2(b) it can be seen that $k_i$ reaches a minimum at $d_1=0$, that is the dielectric layer being air completely. However, as we will see below, in this special case, the backward SPP mode in the lower bulk-mode bandgap of the semiconductor will extend into the upper bulk-mode bandgap, and consequently, the bandwidth of USPP will be reduced. Therefore, we need a trade-off value of $d_1$ to minimize the attenuation coefficient of USPP while maintaining its bandwidth.

Fig. 2. (a) Critical value ($d_c$) of the total thickness of the dielectric layers versus $d_1$. (b) Attenuation coefficient ($k_i$) of USPP versus $d_1$. The loss parameter of the semiconductor is $\nu =0.002\omega _p$. (c) Fraction ($\eta _s$) of the modal energy of USPP distributed in the semiconductor versus $d_1$. In (b) and (c), for each $d_1$, $d_2$ takes its critical value according to Eq. (10), and the frequency of USPP is kept at $f=1.08f_p$. The vertical dotted line indicates the position of $d_1=\lambda _p/75$.

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To solve the above problem, we calculate the dispersion diagrams of SPPs for different dielectric layer structures, and the representative results are displayed in Fig. 3. Figure 3(a) shows the dispersion diagram for $d_1= \lambda _p/150$ and $d_2=0.15\lambda _p$, where $\lambda _p=c/f_p$. It is seen that the SPP mode with $k<0$ in the lower bulk-mode bandgap of the semiconductor extends into the upper bulk-mode bandgap, and as a result, the unidirectional window in the upper bulk-mode bandgap is compressed from the bottom. But our calculations show that such a phenomenon disappears when $d_1\ge \lambda _p/75$, as shown in Fig. 3(b), where $d_1= \lambda _p/75$ and $d_2=0.1\lambda _p$, and the SPP mode with $k<0$ just has an upper-frequency cutoff at the edge of the lower bulk-mode zone. Figure 3(c) shows the dispersion diagram for $d_1= \pi c/2\sqrt {\varepsilon _{r_1}}\omega _b$ and $d_2=0$. In this special case, the upper-frequency cutoff of SPP in the lower bulk-mode bandgap is determined by its asymptotic frequency. In all Figs. 3(a)-3(c), the total thickness of the dielectric layers is taken to be its critical value, i.e., $d=d_1+d_{2c}$, which is closely dependent on $d_1$. As expected, for all three cases, the TIR mode has a lower frequency cutoff very close to the upper limit ($\omega _b$) of the upper bulk-mode bandgap. If $d>d_c$, the lower frequency cutoff of the TIR mode drops below $\omega _b$, as shown in Fig. 3(d), where $d_1= \lambda _p/75$ and $d_2=0.2\lambda _p$. In this case, the unidirectional window in the upper bulk-mode bandgap is compressed from the top.

Fig. 3. Dispersion diagrams for the waveguides with different dielectric heterostructures. (a) $d_1=\lambda _p/150$, $d_2=0.15\lambda _p$; (b) $d_1=\lambda _p/75$, $d_2=0.1\lambda _p$; (c) $d_1=\pi c/(w\sqrt {\varepsilon _{r 1}}\omega _b)$, $d_2=0$; (d) $d_1=\lambda _p/75$, $d_2=0.2\lambda _p$. The solid lines in each figure represent the dispersion curves of bound modes in the waveguide, and the dashed lines in (d) represent light lines in air. The shaded rectangle indicates the unidirectional frequency window.

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By combining all the above results, it is clear that we should choose $d_1= \lambda _p/75$ and $d_2=0.1\lambda _p$ for our structure. In this scenario, the portion of the EM energy of the USPP mode distributed in the dielectric layers reaches a maximum on the premise that the whole upper bulk-mode bandgap is a unidirectional window. From now on, we will restrict our structure with the optimal parameters, i.e., $d_1= \lambda _p/75$ and $d_2=0.1\lambda _p$. For our designed waveguide, attenuation coefficients of the USPP mode in the upper bulk-mode bandgap are plotted in Fig. 4(a), where the corresponding results for the previous structure [Fig. 1(c)] are also included for comparison. The attenuation coefficient of USPP in the present structure is significantly smaller than that for the previous structure. For example, at the midgap frequency $f=1.08f_p$, $k_i=0.0028k_p$ ($k_p=\omega _p/c$) for the present structure and $k_i=0.0118k_p$ for the previous structure, the latter is 4.2 times that of the former. Compared to the previous structure, the fraction of the modal energy of USPP distributed in the cladding layer of the plasmonic platform is substantially increased in the present structure, as illustrated in Fig. 4(b), thereby the absorption effect of the semiconductor on USPP is effectively weakened. Figures 4(c) and 4(d) show the profiles of electric and magnetic fields of the USPP mode at $f=1.08f_p$ for both present and previous structures. The present USPP has field amplitudes large and nearly constant in the cladding layer, while the previous one has field amplitudes rapidly decaying in the cladding. We believe that our structure will also significantly decrease the leakage loss of USPP caused by nonlocality.

Fig. 4. Comparison between the present and previous structures. (a) Attenuation coefficients of USPP in the upper bulk-mode bandgap. The loss parameter of the semiconductor is $\nu =0.002\omega _p$. (b) Fraction of the modal energy of USPP distributed in the cladding layer of the nonreciprocal plasmonic platform. (c,d) Profiles of the electric and magnetic fields of USPP for $f=1.08f_p$. In each figure, the solid line represents the results for the present structure, while the dashed line corresponds to the previous structure.

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3. Characteristic of USPP in the nonlocal model

To strictly reveal the properties of our proposed waveguide, we further investigate the impact of the nonlocal effect of the semiconductor on SPPs in it. Like in [34], we treat the nonlocal effect using a hydrodynamic model of free-electron gas, and the response of the semiconductor to EM field can be represented by an induced free-electron current ($\bf {J}$), which satisfies the equation [28]

(11)$$\beta^2 \nabla(\nabla \cdot {\bf{J}})+\omega(\omega+i \nu) {\bf{J}}+i \omega {\bf{J}} \times \omega_c {\hat{z}}=i \omega \omega_p^2 \varepsilon_0 \varepsilon_{\infty} \bf{E},$$

where $\beta$ is the nonlocal parameter, and $\nu$ is a phenomenological damping rate. Here, we take $\beta =1.07\times 10^6 m/s$ for InSb at room temperature [9]. In the nonlocal model, Maxwell’s equations can be written as $\nabla \times {\bf E}=i\omega \mu _0{\bf H}$ and $\nabla \times {\bf H}=-i\omega \varepsilon _0\varepsilon _{\infty }{\bf H}+{\bf J}$. Due to nonlocality, the magnetized semiconductor is not only temporally dispersive but also becomes spatially dispersive. Correspondingly, the dispersion equation of bulk-modes (for TM polarization) has the form

(12) $$Ak^4+Bk^2+C=0,$$

where $A=\beta ^2 \omega \tilde {\omega }$, $B=\omega ^2 \omega _c^2-(\beta ^2 \varepsilon _{\infty } k_0^2+\omega \tilde {\omega })(\omega \tilde {\omega }-\omega _p^2)$, $C=\varepsilon _{\infty } k_0^2[(\omega \tilde {\omega }-\omega _p^2)^2-\omega ^2 \omega _c^2)]$, where $\tilde {\omega }=\omega +i\nu$. The nonlocal effect enables the semiconductor to support two bulk modes with different lower-frequency cutoffs. We denote the propagation constants of two bulk modes by $k_a$ and $k_b$. The first bulk-mode has a dispersion relation of $k_a^2=(-B+\sqrt {\Delta })/2A$, where $\Delta =B^2-4AC$, and it has a lower frequency cutoff of $\omega _a$. The dispersion relation of the second bulk mode is $k_b^2=(-B-\sqrt {\Delta })/2A$, and its lower frequency cutoff is $\omega _b$. The dispersion curves of both bulk-modes monotonically grow with $k$, and none of them has upper-frequency cutoff, as shown in Fig. 5. Note that for distinguishing propagating modes from evanescent modes, we neglect material dissipation in the mode analysis. In Fig. 5, the dispersion relation for the bulk-mode in the local model is also plotted for comparison. Obviously, the nonlocality induces an additional bulk-mode in the upper bulk-mode bandgap between $\omega _r$ and $\omega _b$, and thus makes it an incomplete bandgap.

Fig. 5. Dispersion diagram of (TM) bulk-modes for the magnetized semiconductor in the nonlocal model. The nonlocal parameter is $\beta =1.07\times 10^6 m/s$. For comparison, the dispersion curves (dotted lines) of bulk modes for the local model are also included. The shaded area indicates the upper bulk-mode bandgap, which becomes incomplete in the nonlocal case.

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We now consider SPPs supported by the upper (incomplete) bulk-mode bandgap of the semiconductor in our structure. As the nonlocal effect makes the semiconductor “birefringent”, the field of SPP has two components in the semiconductor. One is an evanescent wave, and the other is a propagating wave, and the latter causes a leakage loss to SPP. Thus, SPP becomes a leaky mode, and its nonzero field components have the form

(13)$$\begin{aligned}& E_x=\left[A_1 \exp ({-}i p y)+A_2 \exp (\gamma y)\right] \exp (i k x),\\ & E_y=i\left[\sigma_1 A_1 \exp ({-}i p y)+\sigma_2 A_2 \exp (\gamma y)\right] \exp (i k x),\\ & H_z=\frac{i}{\omega \mu_0}\left[\left(\sigma_1 k-i p\right) A_1 \exp ({-}i p y)+\left(\sigma_2 k+\gamma\right) A_2 \exp (\gamma y)\right] \exp (i k x), \end{aligned}$$

with

\begin{aligned}\sigma_1&=\frac{\omega \tilde{\omega} p^2+i \omega \omega_c k p+\varepsilon_{\infty} k_0^2(\beta^2 k^2-\omega \tilde{\omega}+\omega_p^2)}{\omega \omega_c(k^2-\varepsilon_{\infty} k_0^2)-i(\omega \tilde{\omega}- \varepsilon_{\infty}\beta^2 k_0^2) k p}, \\ \sigma_2&=\frac{-\omega \tilde{\omega} \gamma^2-\omega \omega_c k \gamma+\varepsilon_{\infty} k_0^2(\beta^2 k^2-\omega \tilde{\omega}+\omega_p^2)}{\omega \omega_c(k^2-\varepsilon_{\infty} k_0^2)+(\omega \tilde{\omega}- \varepsilon_{\infty}\beta^2 k_0^2) k \gamma}\end{aligned},

where $p=\sqrt {k_a^2-k^2}$ and $\gamma =\sqrt {k^2-k_b^2}$. Because of the leakage effect in the nonlocal semiconductor, SPP supported by the upper bulk-mode bandgap has a nonzero attenuation even in the absence of material loss ($\nu =0$), i.e., the propagation constant is complex, $k=k_r+ik_i$, where $k_r$ and $k_i$ are real-valued. As in the local case, the dispersion equation for SPP in the nonlocal case can be similarly derived by introducing an additional boundary condition, which requires the normal component of ${\bf J}$ to vanish at the interface between the InSb and Si [35]. The dispersion equation of SPP is found to have the following form

(14)$$\begin{aligned} &\varepsilon_{\infty}\left(\sigma_1 \gamma+i p \sigma_2\right) \frac{\alpha_1 \tanh \left(\alpha_1 d_1\right) / \varepsilon_{r 1}+\alpha_2 \tanh \left(\alpha_2 d_2\right) / \varepsilon_{r 2}}{1+\left(\varepsilon_{r 1} \alpha_2 / \varepsilon_{r 2} \alpha_1\right) \tanh \left(\alpha_1 d_1\right) \tanh \left(\alpha_2 d_2\right)}\\ &-k(\gamma+i p)+\left(\sigma_1-\sigma_2\right)\left(k^2-\varepsilon_{\infty} k_0^2\right)=0, \end{aligned}$$

where $\alpha _1=\sqrt {k^2-\varepsilon _{r_1}k_0^2}$ and $\alpha _2=\sqrt {k^2-\varepsilon _{r_2}k_0^2}$.

Figure 6(a) shows the dispersion diagram for our structure in the nonlocal model. Although the nonlocal effect makes the upper bulk-mode bandgap incomplete, it is still able to support USPP, and its dispersion curve (solid line) almost overlaps that (circles) for the local model. However, the nonlocality causes leakage loss to USPP, and thus it becomes a leaky mode. So, even in the case of no material loss, USMP has a nonzero attenuation coefficient, as shown in Fig. 6(b). At the midgap frequency $f=1.08 f_p$, the attenuation coefficient is $k_i=0.008k_p$, which corresponds to a propagation length of nearly $11\lambda$, where $\lambda$ is the wavelength in vacuum. Figure 6(b) also shows the attenuation coefficient (dashed line) of USPP for the previous structure, which is significantly larger than that for the present structure. For example, at $f=1.08 f_p$, the attenuation coefficient is $k_i=0.045k_p$, which is $5.7$ times larger than the result of our structure. This leakage loss corresponds to a propagation length of only $L_p=1.9 \lambda$, which makes previous USPP rather difficult for its practical applications in optical devices. Obviously, our structure effectively solves the issue of leakage loss for actual USMP. Figures 6(c) and 6(d) show the field profiles of the USPP mode for the previous and present structures. In the semiconductor ($y<0$), the fields almost have the same distributions in both the structures, and when $y<-0.3\lambda$, they almost become steady radiating fields. However, in the cladding layer ($y>0$), the fields have completely different distributions for the two structures. The field almost has a uniform amplitude in the dielectric layers for the present structure, while it rapidly decays in the opaque medium for the previous structure. Therefore, compared to the previous structure, the portion of modal energy of USMP in the semiconductor is substantially decreased in the present structure, and this well explains why the leakage loss of USPP is significantly reduced.

Fig. 6. (a) Dispersion curve of USPP in the nonlocal model. The dispersion curve almost overlaps the results (circles) for the local model, indicating that the nonlocal effect does not affect the dispersion feature of USPP. (b) Attenuation coefficient of USPP as a function of frequency for the nonlocal and lossless ($\nu =0$) case. The attenuation coefficients (solid line) for the present structure are significantly smaller than those (dashed line) for the previous structure. (c,d) Profiles of electric and magnetic field amplitudes of USPP for $f=1.08f_p$. The dotted line represents the corresponding results for the previous structure.

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In our structure, however, the nonlocality of the semiconductor not only causes leakage loss to the USPP mode, but also induces an additional SPP mode with $k\ll -k_p$ in the upper bulk-mode bandgap, as shown in Fig. 7(a), which is a more complete dispersion diagram for our structure. This additional SPP mode is backward-propagating, and it seems to be a damage to the unidirectional window. Evidently, the backward SPP results from an extending of the backward SPP mode in the lower bulk-mode bandgap into the upper (incomplete) bulk-mode bandgap, and it has no leakage since it always lies outside the zones of all bulk-modes. But such SPP mode induced by the nonlocality is very sensitive to material loss. Note that the semiconductor is inherently lossy as it is strongly dispersive. Figure 7(b) shows the attenuation coefficients of both forward and backward SPPs in the upper (incomplete) bulk-mode bandgap for a small semiconductor loss of $\nu =0.002\omega _p$. The attenuation of the backward SPP is only caused by the material absorption, while the attenuation of the forward SPP originates from both the material absorption and radiation loss. As seen in the figure, the attenuation coefficient of the backward SPP is much larger than that of the forward one. For example, at the central frequency $f=1.08f_p$, $k_i=0.01k_p$ for the forward SPP and $k_i=-0.588k_p$ for the backward SPP, the latter is nearly $59$ times the former. Though the material loss is very small, the propagation length of the backward SPP is only about $0.1\lambda$ (for $f=1.08f_p$), this is because its field is so tightly confined to the semiconductor-Si interface, as illustrated in Figs. 7(c) and 7(d). As the nonlocality-induced SPP can be suppressed even at a low level of material dissipation, it actually has no damage to the unidirectional behavior of our structure. In Fig. 7(b), the attenuation coefficient (dashed line) of USPP in the previous structure is also plotted (for the nonlocal and lossy case), which is significantly larger than that in the present structure. At $f=1.08 f_p$, the attenuation coefficient is $k_i=0.057k_p$, which is $5.7$ times that of the present structure. On the other hand, in the loss case, the radiation field component of USPP rather quickly decays in the semiconductor and thus distributes only in a very finite (transverse) range, as shown in Figs. 7(c) and 7(d), where the field profiles of USPP are displayed for both the previous and present structures. Evidently, the larger the loss of the semiconductor, the smaller the transverse extent of the USPP field in the semiconductor. Figures 7(c) and 7(d) also explain why USPP in the present structure has a propagation length much larger than that in the previous structure.

Fig. 7. (a) Dispersion curves of SPPs for the waveguide in the nonlocal model. The dotted lines represent the dispersion curves for the two bulk modes in the nonlocal (magnetized) semiconductor. (b) Attenuation coefficient of (actual) USPP as a function of frequency for $\nu =0.002\omega _p$. The dotted line represents the attenuation coefficients of the backward SPP induced by the nonlocality. The dashed line represents the corresponding results of USPP for the previous structure. (c,d) Profiles of the electric and magnetic field amplitudes of USPP at $f=1.08 f_p$ for the nonlocal and lossy case ($\nu =0.002\omega _p$). The dashed line represents the modal field of the backward SPP at the same frequency. The dotted line represents the corresponding modal field of USPP for the previous structure.

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4. Full-wave simulations for unidirectional waveguide

In order to verify the guiding properties of the proposed waveguide, we simulated wave transmission in it for various scenarios using finite element method (FEM). We first performed simulations in the local model. A magnetic current line source was placed at the InSb-Si interface to excite wave, and the frequency is $f=1.08f_p$. The excited wave solely propagates along the positive $x$ direction, as illustrated in Fig. 8(a), which is the simulated magnetic field distribution, and this confirms the unidirectional nature of the waveguide in the local case. To examine the robustness of the USPP, we introduced a defect into the waveguide and resimulated it. The defect consists of a protruding InSb column and a sunken Si column at the InSb-Si interface ($y=0$), both of which have a square cross-section with side length $1$ $\mu$m. But no backward wave is generated by the defect, and the incident SPP completely bypasses it and continues to propagate forward, as illustrated in Fig. 8(b). The defect only modifies the local field around it, and behind it the field of SPP quickly recovers, and this is more clearly displayed in Fig. 8(c), where the distributions of magnetic field amplitude along the interface ($y=0$) are plotted for both cases with and without defect. Clearly, in the local model, our waveguide supports robust USPP, which is consistent with the previous theoretical prediction. In the above simulations, we assumed the semiconductor to be lossless. When the loss of the semiconductor is taken into account, the local electric field generated by the defect might increase the absorption loss in the semiconductor. However, our simulation shows that the power loss caused by the defect is only $0.006{\%}$ for the loss parameter $\nu =0.002\omega _p$, so the impact of the defect on USPP is very weak, as illustrated in Fig. 8(d), which shows the field amplitude distributions along the interface for both lossy cases with and without defect

Fig. 8. Simulations for waveguides in the local model. (a,b) Magnetic field distributions launched by a magnetic current line source at $f=1.08 f_p$ for the cases without and with defect. For both cases, the semiconductor is lossless, i.e., $\nu =0$. The defect in (b) consists of a protruding semiconductor and sunken Si columns, with a square cross-section of side length $1$ $\mu$m. (c) Distributions of magnetic field amplitude along the semiconductor surface ($y = 0$) in (a) and (b). (d) Distributions of magnetic field amplitude similar to those in (c) but for the lossy case with $\nu =0.002\omega _p$.

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Next, we performed simulations of the waveguide in the nonlocal model. Evidently, in the lossless case ($\nu =0$), both forward- and backward-propagating SPPs are launched by the line source, as shown in Fig. 9(a), which is the simulated magnetic field distribution for $f=1.08 f_p$. When the backward SPP reaches the left end of the waveguide, it is partly reflected and then converted into the forward SPP mode. As the field of the backward SPP is so tightly confined to the semiconductor-Si interface, only the forward wave is visible on the left side of the source in the figure. But the backward SPP can be clearly observed from the distribution of the field amplitude along the interface, as illustrated in Fig. 9(c). On the left side of the source, the interference between the backward and forward waves makes field amplitudes be oscillating. However, when a small material loss of $\nu =0.002\omega _p$ is taken into account, no sign of backward wave is again visible [Fig. 9(b)], as it is rapidly attenuated by absorption loss, and this is more clearly illustrated in Fig. 9(c), where the distribution of magnetic field amplitude along the interface is also plotted for the lossy case. It can be seen that the propagation length of the backward mode is much smaller than one wavelength even in this small loss case. All above simulated results are completely consistent with the previous theoretical analysis, and imply that our structure is actually a unidirectional waveguide. Here, we should indicate that since the backward SMP mode is tightly confined to the interface, it is very difficult to excite. If the line source is placed in the middle (i.e., at $1$ $\mu$m above the interface) of the Si layer instead of being at the interface, the backward SPP mode cannot be excited anymore.

Fig. 9. Simulations for waveguides in the nonlocal model. (a,b) Magnetic field distributions at $f=1.08f_p$ for the lossless ($\nu =0$) and lossy ($\nu =0.002\omega _p$) cases. In (b), the backward SPP mode induced by the nonlocality is suppressed by the semiconductor dissipation. (c) Distributions of magnetic field amplitude along the semiconductor surface ($y = 0$) in (a) and (b).

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Interestingly, as in the local case, the (actual) USPP mode in our waveguide is robust to defects in the nonlocal case, as illustrated in Fig. 10, where the same defect as in Fig. 8(b) is introduced into the waveguides with $\nu =0$ and $0.002\omega _p$. No sign of backward SPP is observed even in the lossless case. In Fig. 10, the line source is placed in the middle of the Si layer, so the backward SPP mode cannot be directly excited by it. Evidently, the coupling between the forward and backward SPPs through defect is very weak, because their propagation constants have enormous difference and their modal spots are extremely mismatching [36]. In the lossless case, we evaluated the power between the source and defect, and compared it with the power at the same position in the case without defect, then found that only $0.07{\%}$ of the forward SPP power is coupled into backward SPP by the defect. So the effect of coupling between the forward and backward SPPs is actually negligible. However, the defect might cause non-negligible radiation loss to USPP. Due to the nonlocality, the defect not only generates evanescent field around it as in the local case, but also launches radiation wave (the nonlocality-induced bulk mode) in the semiconductor, and the latter leads to a radiation loss. Note that the nonlocality-induced bulk mode only has a propagation length smaller than one wavelength even at low level of material dissipation. The evanescent field (around the defect) in the nonlocal case should almost be identical to that in the local case, so its absorption loss in the semiconductor is also negligible. In the lossless case, it is difficult to evaluate the radiation loss by the defect, as the radiation field partially overlaps the USPP field. Thus, in the lossy case ($\nu =0.002\omega _p$), the power at a certain distance behind the defect is calculated, and compared with the corresponding power in the case without defect. The total power loss caused by the defect is $1.78{\%}$, which is much larger than both the SPP coupling loss and evanescent-field absorption loss. Clearly, the defect actually leads to three kinds of losses, i.e., SPP coupling loss, evanescent-field absorption loss, and radiation loss, and the first two are negligible, but the last one may be observable.

Fig. 10. Simulations for waveguides with defect in the nonlocal model. (a,b) Magnetic field distributions at $f=1.08 f_p$ for the lossless ($\nu =0$) and lossy ($\nu =0.002\omega _p$) cases. The defect is the same as in Fig. 8(b). (c,d) Distributions of magnetic field amplitude along the semiconductor surface ($y=0$) in (a) and (b), respectively. The corresponding results (dashed line) for the case without defect are included for comparison.

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We think that the radiation loss from defect mainly originates from the scattering of the radiation field component of the USPP mode, hence it is closely dependent on the shape of defect. Moreover, for the USPP mode, its radiation field component should be easier to be strongly scattered than its evanescent field component. To clarify this issue, we further performed simulations for two cases with different defects in the nonlocal and lossy ($\nu =0.002\omega _p$) model. Here, for convenience, the previous defect is denoted as type-I defect, and the present two defects as type-II and type-III defects, which are InSb and Si columns, respectively. The type-II defect lies on the semiconductor surface, whereas the type-III defect is under the semiconductor surface. Both defects have a square cross-section of side length $1$ $\mu$m. For the type-II defect, the total power loss caused by it is only $0.006{\%}$, which is nearly equal to the value obtained from the lossy local model. This implies that this defect located outside the semiconductor only has a very weak influence on the USPP, as it cannot directly act on the radiation field component of the USPP. For the type-III defect, the total power loss becomes $1.64{\%}$, which is very close to the value ($1.78{\%}$) of the type-I defect. In the lossy local model, it’s worth noting that this defect almost causes no power loss to the USPP, primarily because the generated evanescent field mainly distributes in the Si medium. Clearly, in comparison to the type-II defect situated on the semiconductor surface, the type-III defect embedded within the semiconductor can effectively scatter the radiation field component of the USPP, resulting in a comparatively larger power loss. Correspondingly, this leads to substantial modifications in the field around the type-III defect, as depicted in Fig. 11, which shows the distributions of magnetic field amplitude along the interface ($y=0$) for the three defect cases. The local field around the type-III defect is almost identical to that around the type-I defect, which is a combination of the type-II and type-III defects, and this is consistent with the fact that the field is only slightly modified by the type-II defect. The obtained results provide some implications for actually fabricating high-quality plasmonic waveguides.

Fig. 11. (a) Distributions of magnetic field amplitude along the semiconductor surface at $f=1.08f_p$ for the nonlocal and lossy waveguides with different defects. The loss parameter is $\nu =0.002\omega _p$. (b) Zoomed view of local fields around the defects in (a).

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5. Conclusion

In summary, we have conducted a comprehensive analysis of the guiding properties of a semiconductor-Si-air-metal layered structure under an external magnetic field. Our findings demonstrate that, by properly choosing the structural parameters, a sole unidirectional SPP mode exclusively resides in the upper bulk-mode bandgap of the magnetized semiconductor. This mode exists for small wavenumbers in the whole bandgap, hence the nonlocal effect does not affect its dispersion feature over a broad band, i.e., the whole bandgap is a robust unidirectional window. Compared to previous USPPs existing at the interface between a magnetized semiconductor and an opaque isotropic medium, the leakage loss of the present USPPs, which is caused by nonlocality, is largely decreased by more than five times, since its modal energy has a substantially large portion distributed in the cladding layer of the nonreciprocal plasmonic platform. The situation for its absorption loss (caused by the semiconductor dissipation) is almost the same. Though a backward-propagating SPP mode is also induced by the nonlocality in the upper bulk-mode bandgap, which corresponds to an upward expansion of a conventional surface magnetoplasmon mode in the lower bulk-mode bandgap, it can be suppressed at a very low level of material dissipation because of its deep-subwavelength confinement to the interface between the semiconductor and Si. Therefore, our proposed waveguide actually exhibits excellent unidirectionality. Besides, our results have also shown that the effect of coupling between the actually unidirectional and nonlocality-induced backward SPPs (through defects) is very weak, since their propagation constants and modal spots are all extremely mismatching. Hence, our USPP is rather robust to nonlocality. We expect that our work will stimulate intensive research to discover new classes of USPPs with excellent properties and enable their practical applications in optical devices.

Funding

National Natural Science Foundation of China (62075197); Natural Science Foundation of Zhejiang Province (Z22F047705).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data Availability

No data were generated or analyzed in the presented research.

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Broadband unidirectional surface plasmon polaritons with low loss

Abstract

1. Introduction

2. Characteristic of USPP in the local model

3. Characteristic of USPP in the nonlocal model

4. Full-wave simulations for unidirectional waveguide

5. Conclusion

Funding

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (11)

Equations (17)

Optics Express