Chiral quantum optics [1] has been established and developed rapidly in recent years. It begins with the study of a novel spin–orbit coupling of photons in strong transversely confined light field. Rich controllable degrees of freedom of photons [2] enable a variety of spin–orbit coupling interactions [3], of which an important effect is called chiral coupling. Due to the considerable longitudinal (along the propagation direction) component of the electric field in the region of the strong transversely confined field, an extraordinary transversely circularly polarized state of photons will be generated. This is called transverse spin (T-spin) [4–9], where the angular-momentum direction of the electric field rotation (i.e., spin direction) is perpendicular to the propagation direction of light. T-spin is usually spatially localized, with a prototypical feature that the spin direction is locked to the propagation direction of the waveguide mode, i.e., the spin-momentum locking effect [9–12], or the quantum spin Hall effect of photons [13]. T-spin also exists in some special free-space light field [14–17] and near field of nanoparticles [18,19]. Bulk modes with global T-spin have been constructed with sophisticated inversely designed metamaterials [20,21]. Recently, the research of T-spin has been extended to topological physics [22,23], optical forces [24–26], exceptional point [27], sound field [28], etc. Apart from the research significance of T-spin, it has inspired the exploration of many applications, of which the most striking one is the chiral coupling between a circularly polarized emitter and waveguide modes. Relying on the spin-momentum locking, the stationary matter qubits (spins) carried in the emission source can be read out deterministically and converted into flying photonic qubits (waveguide modes) for remote information exchange. Based on the platforms of nanofiber waveguides [29–35], photonic crystal waveguides [36–43], and dielectric nanobeam waveguides [44–48], T-spin and chiral-coupling effect have been applied to exploit a variety of functional devices, including quantum information network nodes [39], quantum gates [42], quantum entanglements [38], nanophotonic non-reciprocal [49] devices of isolators [32,50] and circulators [34] by using the spin-polarized atoms or quantum dots as non-reciprocal absorbers, and are expected to play an important role in on-chip integrated photonic circuits and quantum information processing.

Surface plasmon polaritons (SPPs) can confine the electromagnetic field down to deep subwavelength scale with significant local field enhancement [51,52]. It can be utilized to facilitate on-chip integrated and more miniaturized photonic devices [53–56]. With the chiral coupling between emission sources and waveguide modes supported in metal nanowires, the deterministic readout [57] and initialization [58] of valley degrees of freedom in two-dimensional materials, chiral Raman signal detection [59], and on-chip chiral material sensing [60] have been studied on the platform of SPPs.

In addition to the direct chiral coupling between sources and waveguide modes, the indirect chiral coupling can be achieved by introducing the resonant modes as coupling intermediaries. Compared with the waveguide modes, the resonant modes provide stronger chiral field, which can significantly enhance the coupling between the chiral field and emitters [43,50,61], absorbers [32,34,50], or scatterers [27], and can simultaneously enhance the spontaneous emission rate of the chiral source [43,62], or the loss rate of unidirectional waveguide modes caused by chiral absorbers [32,34]. So far, the whispering gallery mode (WGM) supported in the dielectric microcavity has been introduced into nanofiber waveguide systems [32,34,63] and silicon waveguide systems [50] to implement indirect chiral coupling. The chiral coupling between the source and WGMs is similar to that between the source and waveguide modes [61], which originates from the fact that these two degenerate counter-propagating WGMs have opposite chiralities of T-spin at the position of the source. In addition, by introducing a metallic nanoparticle into a photonic crystal waveguide [43] or a metallic nanoblock into a dielectric nanowire waveguide [62], localized surface plasmon resonance (LSPR) modes have been exploited to improve the chiral-coupling rate and the spontaneous emission rate of the chiral source. However, these indirect chiral-coupling systems with LSPRs are still restricted by the fact that the sources must be in the limited evanescent-field region of waveguide modes (similar to the case of direct chiral-coupling systems), and are difficult to fabricate due to the sophisticated design of the structures. Furthermore, indirect chiral-coupling systems more compact than the WGM microcavity are still lacking and need to be explored to meet the requirement of miniaturization in integrated photonic circuits.

Concerning the theories for analyzing the chiral coupling from an emitter to waveguide modes, the waveguide mode expansion of Green’s function (WME-GF) [36,38,64,65] or its equivalent forms [10,11,36,37,44,64] can provide an analytical dependence of the coupling rate on the position and polarization of the emitter and on the electromagnetic field of the waveguide modes, and has been a commonly used approach to achieve an intuitive explanation and quantitative calculation of the chiral-coupling rate. The WME-GF is applicable to the direct chiral-coupling system (i.e., a waveguide without additional coupling structures that may support resonant modes), and its complex-conjugate form is rigorous for lossless waveguide modes (see details in Section 2). The chiral-coupling rate can be also calculated exactly by extracting the waveguide mode coefficients from the total field with the mode orthogonality theorem [48] or calculated approximately by calculating the power of the total field on the waveguide cross section far away from the emitter [43,57,62,66], which is performed in a fully numerical way without an intuitive analyticity. The angular spectrum approach [12,67] has been employed to provide an intuitive explanation of the chiral coupling but not for a quantitative calculation of the chiral-coupling rate. The quasinormal mode (QNM) expansion theory is used for analyzing the chiral coupling between the source and WGMs [61], which are resonant eigenmodes at complex eigenfrequencies [68–71] different from waveguide modes at real frequencies. Presently, there is still a lack of intuitive and quantitative theoretical approaches for analyzing the indirect chiral coupling between the emission source and waveguide modes mediated by resonant modes.

In this paper, we first present a general approach based on the reciprocity theorem for an intuitive analysis and rigorous calculation of the chiral-coupling coefficient ($\beta$) from the source to the waveguide mode (Section 2). With this approach, we derive the conditions for the occurrence of chiral coupling, and some general relations between the chiral-coupling directionality ($g$) and the T-spin ($S_3$) of the field or source. We emphasize that these theories are generally applicable to any direct/indirect chiral-coupling systems and to any lossless/lossy waveguide modes.

Based on the theories, we propose to use the Fabry–Perot (FP) resonant mode formed by dual SPP modes as a novel way to achieve the indirect chiral coupling between the source and waveguide mode (Section 3). This way gets rid of the restriction that the source must be located in the limited evanescent-field region of the waveguide mode for the direct chiral-coupling system, which is a merit similar to that of the indirect chiral coupling mediated by the WGM microcavity [32,34,50,63], and additionally benefits from the deep-subwavelength footprint of the SPP FP-nanocavity. The different orders of FP resonance provide new and flexible design freedoms to control the directionality ($g$) of chiral coupling and can simultaneously achieve an enhancement of the chiral-coupling rate (${\mathrm{\Gamma }}_{\mathrm{SPP},\pm }$) of the SPP waveguide mode, the total spontaneous emission rate (${\mathrm{\Gamma }}_{\mathrm{total}}$) of the source, and the chiral-coupling efficiency (${\eta }_{\pm }={\mathrm{\Gamma }}_{\mathrm{SPP},\pm }/{\mathrm{\Gamma }}_{\mathrm{total}}$). Therefore, the proposed system is expected to attain a fast and deterministic readout of the spins of the source on a more compact and flexibly designed chip. Two first-principles-based SPP models are built up for the reciprocal and original problems, respectively, which quantitatively verify the decisive role of the FP resonance in achieving the chiral coupling and the simultaneous enhancement of the spontaneous emission rate.

In this section, we will present a general approach based on the reciprocity theorem for an intuitive analysis and rigorous calculation of the direct or indirect chiral-coupling coefficients from the source to waveguide modes. For the coupling system, a point source located at ${\mathbf{r}}_0=(x_0,y_0,z_0)$ can be expressed as a current density $\mathbf{j}=\delta (\mathbf{r}-{\mathbf{r}}_0)\mathbf{p}$. According to the reciprocity theorem [72,73], the coefficients of the $m$th waveguide mode excited by the point source can be expressed as $${\beta }_{+,m}=\frac{{\mathbf{E}}_{-,m}^{\text{coupling}}({\mathbf{r}}_0)·\mathbf{p}}{⟨{\mathit{\psi }}_{-,m}|{\mathit{\psi }}_{+,m}⟩},{\beta }_{-,m}=\frac{{\mathbf{E}}_{+,m}^{\text{coupling}}({\mathbf{r}}_0)·\mathbf{p}}{⟨{\mathit{\psi }}_{-,m}|{\mathit{\psi }}_{+,m}⟩},$$(1)where $\mathit{\psi }=[\mathbf{E},\mathbf{H}]$ denotes both the electric field $\mathbf{E}$ and the magnetic field $\mathbf{H}$. ${\mathit{\psi }}_{+,m}$ and ${\mathit{\psi }}_{-,m}$ represent the $m$th waveguide modes with unitary coefficients (or called normalized modes) propagating along positive and negative directions of the $z$ axis, respectively, with the $z$ axis set along the translationally invariant direction of the waveguide. ${\mathbf{E}}_{+,m}^{\text{coupling}}({\mathbf{r}}_0)$ or ${\mathbf{E}}_{-,m}^{\text{coupling}}({\mathbf{r}}_0)$ denotes the electric field at ${\mathbf{r}}_0$ excited by an incident ${\mathit{\psi }}_{+,m}$ or ${\mathit{\psi }}_{-,m}$, respectively, and corresponds to the reciprocal problem for which the point source is absent [as shown by Fig. 1(b)]. The problem with the excitation of the point source is called the original problem in this paper [as shown by Fig. 1(a)]. In Eq. (1), a bilinear function is defined as $$⟨{\mathit{\psi }}_1|{\mathit{\psi }}_2⟩={\iint }_{z=z_1}({\mathbf{E}}_2\times {\mathbf{H}}_1-{\mathbf{E}}_1\times {\mathbf{H}}_2)·\mathbf{z}\mathrm{d}x\mathrm{d}y,$$(2)where $z_1$ can be selected arbitrarily. Equation (1) is generally applicable to any direct/indirect coupling systems (i.e., waveguide without/with additional coupling structures that may support resonant modes), in which the polarization of the source can be arbitrary. In particular, in direct coupling systems, ${\mathbf{E}}_{\pm ,m}^{\text{coupling}}({\mathbf{r}}_0)={\mathbf{E}}_{\pm ,m}({\mathbf{r}}_0)$, reducing to the electric field of waveguide mode at ${\mathbf{r}}_0$.

Now we consider the calculation of waveguide mode coefficients excited by a circularly polarized point source based on Eq. (1). The right-handed circularly polarized point source can be expressed as $\mathbf{j}={\mathbf{j}}_{\sigma }=\delta (\mathbf{r}-{\mathbf{r}}_0){\mathbf{p}}_{\sigma }$, where ${\mathbf{p}}_{\sigma }=\mathbf{x}+\mathbf{z}i$, with $\mathbf{x}$ and $\mathbf{z}$ being unit vectors along the $x$ and $z$ directions, respectively. ${\mathbf{j}}_{\sigma }$ can be decomposed into a superposition of two linearly polarized point sources ${\mathbf{j}}_x=\delta (\mathbf{r}-{\mathbf{r}}_0)\mathbf{x}$ and ${\mathbf{j}}_z=\delta (\mathbf{r}-{\mathbf{r}}_0)\mathbf{z}$ with orthogonal polarization directions and a phase difference of $\pi /2$, i.e., ${\mathbf{j}}_{\sigma }={\mathbf{j}}_x+{\mathbf{j}}_{z}i$. In the following discussion, it is assumed that the coupling structure is symmetric with respect to the $z=z_0$ plane where the source is located (as sketched in Fig. 1). According to the symmetry of the electromagnetic field [73], it can be concluded that the coefficients of the $m$th forward- and backward-propagating waveguide modes excited by ${\mathbf{j}}_x$ are equal, which are denoted by ${\beta }_m^{\perp }$; while those excited by ${\mathbf{j}}_z$ are opposite, which are denoted by ${\beta }_m^{||}$ and $-{\beta }_m^{||}$, respectively. According to the principle of linear superposition of the electromagnetic field, $${\beta }_{+,m}={\beta }_m^{\perp }+i{\beta }_m^{||}=\frac{E_{x,-,m}^{\text{coupling}}({\mathbf{r}}_0)+i{E}_{z,-,m}^{\text{coupling}}({\mathbf{r}}_0)}{⟨{\mathit{\psi }}_{-,m}|{\mathit{\psi }}_{+,m}⟩},$$(3a)$${\beta }_{-,m}={\beta }_m^{\perp }-i{\beta }_m^{||}=\frac{E_{x,+,m}^{\text{coupling}}({\mathbf{r}}_0)+i{E}_{z,+,m}^{\text{coupling}}({\mathbf{r}}_0)}{⟨{\mathit{\psi }}_{-,m}|{\mathit{\psi }}_{+,m}⟩},$$(3b)where the second equalities are obtained with the use of Eq. (1), and the subscripts $x$ and $z$ denote the $x$ and $z$ components of the electric field vector $\mathbf{E}$. The second equalities of Eq. (3) further yield $$E_{x,-,m}^{\text{coupling}}({\mathbf{r}}_0)=E_{x,+,m}^{\text{coupling}}({\mathbf{r}}_0),-E_{z,-,m}^{\text{coupling}}({\mathbf{r}}_0)=E_{z,+,m}^{\text{coupling}}({\mathbf{r}}_0).$$(4)Note that Eq. (4) can be also consistently derived from the symmetry of the electromagnetic fields excited by the forward- and backward-propagating waveguide modes.

Next, based on Eq. (3), it is possible to explicitly demonstrate the conditions for the occurrence of chiral coupling between the circularly polarized point source and waveguide modes. Equation (3) indicates that if $$E_{x,-,m}^{\text{coupling}}({\mathbf{r}}_0)=-i{E}_{z,-,m}^{\text{coupling}}({\mathbf{r}}_0)\iff E_{x,+,m}^{\text{coupling}}({\mathbf{r}}_0)=i{E}_{z,+,m}^{\text{coupling}}({\mathbf{r}}_0),$$(5)i.e., the T-spin of ${\mathbf{E}}_{-,m}^{\text{coupling}}({\mathbf{r}}_0)$ is consistent with the spin of ${\mathbf{j}}_{\sigma }$ (right-handed circular polarization), or equivalently, the T-spin of ${\mathbf{E}}_{+,m}^{\text{coupling}}({\mathbf{r}}_0)$ is contrary to the spin of ${\mathbf{j}}_{\sigma }$, there will be ${\beta }_{+,m}=0$ and ${\beta }_{-,m}\ne 0$, which means the occurrence of chiral coupling that only ${\mathit{\psi }}_{-,m}$ is excited by ${\mathbf{j}}_{\sigma }$. Similarly, if $$E_{x,+,m}^{\text{coupling}}({\mathbf{r}}_0)=-i{E}_{z,+,m}^{\text{coupling}}({\mathbf{r}}_0)\iff E_{x,-,m}^{\text{coupling}}({\mathbf{r}}_0)=i{E}_{z,-,m}^{\text{coupling}}({\mathbf{r}}_0),$$(6)i.e., the T-spin of ${\mathbf{E}}_{+,m}^{\text{coupling}}({\mathbf{r}}_0)$ is consistent with the spin of ${\mathbf{j}}_{\sigma }$, or equivalently, the T-spin of ${\mathbf{E}}_{-,m}^{\text{coupling}}({\mathbf{r}}_0)$ is contrary to the spin of ${\mathbf{j}}_{\sigma }$, there will be ${\beta }_{-,m}=0$ and ${\beta }_{+,m}\ne 0$, which means the occurrence of chiral coupling that only ${\mathit{\psi }}_{+,m}$ is excited by ${\mathbf{j}}_{\sigma }$.

The above analysis shows that there exists a locking effect between the propagation direction (momentum) of the waveguide mode ${\mathit{\psi }}_{\pm ,m}$ and the transverse circular polarization (T-spin) of the electric field ${\mathbf{E}}_{\pm ,m}^{\text{coupling}}({\mathbf{r}}_0)$ excited by ${\mathit{\psi }}_{\pm ,m}$ [as shown in Eqs. (5) and (6)]. Accordingly, based on Eq. (3) derived from the reciprocity theorem, there will exist a locking effect between the circular polarization (spin) of the chiral source ${\mathbf{j}}_{\sigma }$ and the propagation direction (momentum) of the waveguide mode excited by ${\mathbf{j}}_{\sigma }$.

Based on the above analysis, we will further derive some novel general relations between the directivity of the excited waveguide mode and the T-spin of the field or point source, as stated in the following.

A directivity factor $g$ of the excited waveguide mode ${\mathit{\psi }}_{\pm ,m}$ can be defined as $g=(|{\beta }_{+,m}{|}^2-|{\beta }_{-,m}{|}^2)/(|{\beta }_{+,m}{|}^2+|{\beta }_{-,m}{|}^2)$, where $g=1$ or $-1$ means a perfect unidirectional excitation of ${\mathit{\psi }}_{+,m}$ or ${\mathit{\psi }}_{-,m}$, respectively. The T-spin of ${\mathbf{E}}_{+,m}^{\text{coupling}}({\mathbf{r}}_0)$ can be described by the Stokes parameter [5,7,44,57], $S_3({\mathbf{r}}_0)=\frac{-2\mathrm{Im}[E_{x,+,m}^{\text{coupling}}({\mathbf{r}}_0)E_{z,+,m}^{\text{coupling}}{({\mathbf{r}}_0)}^{*}]}{{|E_{x,+,m}^{\text{coupling}}({\mathbf{r}}_0)|}^2+{|E_{z,+,m}^{\text{coupling}}({\mathbf{r}}_0)|}^2}$, where $S_3({\mathbf{r}}_0)=-1$, 0, 1 correspond to left-handed, null, and right-handed T-spin of ${\mathbf{E}}_{+,m}^{\text{coupling}}({\mathbf{r}}_0)$, respectively. Then from Eqs. (3) and (4), which are derived under the two assumptions that the point source is right-handed circularly polarized and that the coupling structure is symmetric with respect to the $z=z_0$ plane, it can be proved that the $g$ defined for the original problem and $S_3$ defined for the reciprocal problem satisfy a general relation (see Appendix A.2): $$g=S_3({\mathbf{r}}_0).$$(7)Equation (7) can be regarded as a generalization of Eqs. (5) and (6) for the special cases of $g=-1$ and 1 to the general case of $-1\le g\le 1$.

Furthermore, we consider a point source $\mathbf{j}=\delta (\mathbf{r}-{\mathbf{r}}_0)\mathbf{p}$ with an arbitrary polarization, i.e., $\mathbf{p}=p_x\mathbf{x}+p_z\mathbf{z}$, and assume that the coupling structure is designed to satisfy Eq. (6), i.e., $S_3({\mathbf{r}}_0)=1$. Then substituting $\mathbf{p}=p_x\mathbf{x}+p_z\mathbf{z}$ into Eq. (1) and using Eqs. (4) and (6), one can derive another general relation (see Appendix A.3): $$g=\frac{-2\mathrm{Im}(p_x{p}_z^{*})}{{|p_x|}^2+{|p_z|}^2}.$$(8)The right side of Eq. (8) is defined as the $S_3$ parameter of the point source, denoted by $S_{3,\mathrm{source}.}{S}_{3,\mathrm{source}}=-1$, 0, 1 correspond to left-handed circularly polarized, linearly polarized, and right-handed circularly polarized point sources, respectively, and according to Eq. (8), will lead to a unidirectional excitation of ${\mathit{\psi }}_{-,m}$, a bidirectional excitation of both ${\mathit{\psi }}_{-,m}$ and ${\mathit{\psi }}_{+,m}$, and a unidirectional excitation of ${\mathit{\psi }}_{+,m}$, respectively.

Note that for Eq. (7), a change of $S_3({\mathbf{r}}_0)$ or $g$ implies a change of the coupling structure while the polarization of the point source remains unchanged to be right-handed circular polarization. For Eq. (8), differently, a change of $S_{3,\mathrm{source}}$ or $g$ implies a change of the polarization of the source while the coupling structure remains unchanged. Therefore, the significance of Eq. (7) is that it provides a general tool for designing the structure [through designing the $S_3({\mathbf{r}}_0)$] to achieve the desired chiral coupling ($g$); while the significance of Eq. (8) lies in the fact that it explicitly demonstrates the feasibility to read out the static qubit of the source ($S_{3,\mathrm{source}}$) by measuring the flying qubit of photons ($g$) [1,46].

Here we emphasize that the above proposed theories are generally applicable to any direct/indirect chiral-coupling systems (i.e., waveguide without/with additional coupling structures that may support resonant modes) and to any lossless/lossy waveguide modes (i.e., propagation constants being real/complex), and can be readily extended to $z$-periodic waveguides [72] such as the photonic crystal waveguides widely used for the chiral coupling [36–43]. The WME-GF with a complex-conjugate form commonly used in the literature [10,11,36–38,44,64,65] can be regarded as a special case of our theories for direct chiral-coupling systems and lossless waveguide modes. To see the point, in Appendix A.1, we provide a derivation of a general WME-GF for lossy waveguide modes based on the QNM expansion formalism [68–71], which explicitly shows how the general WME-GF reduces to the complex-conjugate WME-GF for lossless waveguide modes. A numerical example (Fig. 8 in Appendix A.1) is provided to show the higher accuracy of the general WME-GF compared with the complex-conjugate WME-GF for lossy waveguide modes. This derivation can be regarded as an extension of the derivation of the complex-conjugate WME-GF in the literature (see Eq. (22) in Ref. [65] or Eq. (47) in Ref. [64], for instance) based on the normal-mode expansion formalism. Furthermore, the derived general WME-GF is consistent with our theories derived from the reciprocity theorem, and thus provides a logic cross-check of our theories.

Based on Eqs. (5) and (6) [or, the general relation of Eq. (7)], we then design an indirect chiral-coupling system mediated by an FP resonance of dual SPP modes on the platform of an SPP nanowire waveguide. As shown in Fig. 2(a), the system consists of three parts: a nanowire SPP waveguide, a single-wire antenna (SA), and a double-wire antenna (DA). A circularly polarized point source is located in the gap between the SA and the right terminal of the DA. The material of the structure is gold, with wavelength-dependent refractive index $n_m$ taking tabulated values from Ref. [74]. For the emission wavelength $\lambda =750\mathrm{nm}$ of the source which is considered throughout this paper, there is $n_m=0.1681+4.584i$. The structure is in the air environment without a substrate to simplify the analysis. All the cross sections of nanowires are square with a side length of $D=40\mathrm{nm}$. The widths of the air gap in the DA and of the gap between the SA and the right terminal of the DA are all $G=20\mathrm{nm}$. Note that high-definition ultrasmooth gold nanostructures such as nanowires with widths and gaps down to tens of nanometers are achievable experimentally [75].

Because the size of the cross sections of the nanowires is much smaller than the wavelength, only the fundamental SPP modes of the waveguide, SA, and DA, are bounded (field decaying to null at infinity in transversal directions) and propagative (propagation constant being almost real). For the designed system, only the bounded and propagative SPP modes are needed to be considered. The reason is that with regard to the formation of the chiral field at the position of the source, the contributions of other unbounded or evanescent modes can be neglected, which can be quantitatively verified by the SPP models built up later. The waveguide and the SA only support one fundamental SPP mode, whose electric field distribution is shown in Fig. 2(b1). While the DA supports two fundamental SPP modes, whose electric-field vectors are mirror antisymmetric and symmetric with respect to $z=0$ plane [the coordinate system shown in Fig. 2(a)], as shown in Figs. 2(b2) and 2(b3), respectively.

From Figs. 2(b2) and 2(b3), one can see that the antisymmetric and symmetric SPPs provide dominant electric-field components $E_z$ and $E_x$ in the gap of the DA, respectively, which are the very two field components to excite the T-spin of the field at the position of the source [see the right side of Eq. (3)]. The incident fundamental SPP mode on the waveguide will excite the antisymmetric and symmetric SPPs on the DA simultaneously. However, compared with the antisymmetric SPP, the FP resonance of the symmetric SPP on the DA is weaker because of the weaker confinement of the field, which results in a weaker $E_x$ than $E_z$. To overcome this difficulty, we put the SA close to the right terminal of the DA. Thus, when the SA is at an FP resonance, the $E_x$ in the gap between the DA and the SA will be enhanced significantly, so that a chiral electric field can be constructed. Here, the length of the SA is set as $L_{\mathrm{SA}}=370\mathrm{nm}$, making the SA at a lower order of FP resonance (for which the $L_{\mathrm{SA}}$ is smaller and the resonance is stronger). The calculation process to determine the $L_{\mathrm{SA}}$ is provided in Appendix B.1.

Figure 2(c) shows that for specific lengths of the DA, an incident forward-propagating SPP on the waveguide can excite the FP resonances of both the DA and the SA [the field distribution at the FP resonance of the SA is shown in Fig. 9(b) of Appendix B.1], which results in an enhanced electric field in the gap between the DA and the SA. Therefore, it is expected that if a circularly polarized source is placed in the gap between the DA and the SA [as shown by the red dot in Fig. 2(a)], an indirect chiral coupling between the source and the SPP on the waveguide is likely to be achieved. A detailed analysis will be provided in the next subsection.

For the proposed chiral-coupling system, we will build up an SPP model for the reciprocal problem in which an incident SPP waveguide mode is considered based on Eq. (3), so as to provide a quantitative analysis of the physical mechanism of the chiral coupling. The model is based on a multiple-scattering process of the fundamental SPPs on the DA, in which other non-bounded or non-propagative higher-order modes are neglected, and can provide analytical expressions of the coefficients of the SPP on the waveguide excited by a circularly polarized source. All the parameters in the model are obtained with the first-principles calculations without fitting the numerical results or experimental data, which ensures a solid electromagnetic foundation and thus a quantitative prediction of the model.

According to Eq. (3), for the calculation of the coefficients ${\beta }_{+}$ and ${\beta }_{-}$ of the forward and backward fundamental SPPs on the waveguide excited by a circularly polarized source [as shown in Fig. 2(a)], the corresponding reciprocal problems are for the calculation of the electric field at the position of the source excited by an incident backward or forward fundamental SPP on the waveguide, respectively. The incident forward/up-going SPP from the bottom of the waveguide will excite the right-going and left-going symmetric (antisymmetric) SPPs on the DA with coefficients $a_{+}^{\mathrm{sym}}$ and $b_{+}^{\mathrm{sym}}$ ($a_{+}^{\mathrm{asym}}$ and $b_{+}^{\mathrm{asym}}$), respectively, as shown in Fig. 3(a1). Here all the SPPs with unitary coefficients satisfy the normalizations (on the cross sections where the SPPs begin to propagate) as specified in Fig. 2(b). To determine the SPPs’ coefficients, a set of coupled-SPP equations can be written: $$a_{+}^{\mathrm{sym}}={\alpha }_{+}^{\mathrm{sym}}+b_{+}^{\mathrm{sym}}{u}^{\mathrm{sym}}{r}_b^{\mathrm{sym}},$$(9a)$$b_{+}^{\mathrm{sym}}=a_{+}^{\mathrm{sym}}{u}^{\mathrm{sym}}{r}_a^{\mathrm{sym}},$$(9b)$$a_{+}^{\mathrm{asym}}={\alpha }_{+}^{\mathrm{asym}}+b_{+}^{\mathrm{asym}}{u}^{\mathrm{asym}}{r}_b^{\mathrm{asym}},$$(9c)$$b_{+}^{\mathrm{asym}}=a_{+}^{\mathrm{asym}}{u}^{\mathrm{asym}}{r}_a^{\mathrm{asym}},$$(9d)where $u^{\mathrm{sym}}=\exp (i{k}_0{n}_{\mathrm{eff}}^{\mathrm{sym}}L)$ and $u^{\mathrm{asym}}=\exp (i{k}_0{n}_{\mathrm{eff}}^{\mathrm{asym}}L)$ are the phase-shift factors of the symmetric and antisymmetric SPPs accumulated over the DA, respectively, with $k_0=2\pi /\lambda$ being the wavenumber in vacuum, $L$ being the length of the DA, and $n_{\mathrm{eff}}^{\mathrm{sym}}$ and $n_{\mathrm{eff}}^{\mathrm{asym}}$ being the complex effective indices of the symmetric and antisymmetric SPPs, respectively, which are obtained with a fully vectorial aperiodic Fourier modal method (a-FMM) [76,77]. ${\alpha }_{+}^{\mathrm{sym}}$ (${\alpha }_{+}^{\mathrm{asym}}$) is the coefficient of the right-going symmetric (antisymmetric) SPP on a semi-infinite DA excited by the up-going SPP on the waveguide, as shown in Fig. 3(a2). $r_a^{\mathrm{sym}}$ ($r_a^{\mathrm{asym}}$) is the reflection coefficient of a right-going incident symmetric (antisymmetric) SPP at the right terminal of a semi-infinite DA coupled with the SA, as shown in Fig. 3(a3). $r_b^{\mathrm{sym}}$ ($r_b^{\mathrm{asym}}$) is the reflection coefficient of a left-going incident symmetric (antisymmetric) SPP at the left terminal of a semi-infinite DA coupled with the waveguide, as shown in Fig. 3(a4). Since the symmetric and antisymmetric SPPs on the DA cannot excite each other, Eqs. (9a), (9b) and Eqs. (9c), (9d) are decoupled and can be solved independently. Here $r_a^{\text{sym/asym}}$ and $r_b^{\text{sym/asym}}$ can be calculated as the scattering matrix elements [78] with the fully vectorial a-FMM. ${\alpha }_{+}^{\text{sym/asym}}$ can be calculated with the mode orthogonality theorem [73,79]: $$a_{+}^{\text{sym/asym}}=\frac{⟨{\mathit{\psi }}_{-}^{\text{sym/asym}}|{\mathit{\psi }}_{+,\text{total}}⟩}{⟨{\mathit{\psi }}_{-}^{\text{sym/asym}}|{\mathit{\psi }}_{+}^{\text{sym/asym}}⟩},$$(10)where ${\mathit{\psi }}_{+,\mathrm{total}}$ is the total electromagnetic field in the structure as shown in Fig. 3(a2) excited by the up-going SPP on the waveguide, ${\mathit{\psi }}_{+}^{\mathrm{sym}}$ and ${\mathit{\psi }}_{+}^{\mathrm{asym}}$ (${\mathit{\psi }}_{-}^{\mathrm{sym}}$ and ${\mathit{\psi }}_{-}^{\mathrm{asym}}$) are the electromagnetic field of the right-going (left-going) symmetric and antisymmetric fundamental SPP modes on the DA, respectively. ${\mathit{\psi }}_{+,\mathrm{total}}$ and ${\mathit{\psi }}_{+/-}^{\text{sym/asym}}$ are all calculated with the a-FMM. The definition of the bilinear function $⟨{\mathit{\psi }}_1|{\mathit{\psi }}_2⟩$ is given by Eq. (2), with a replacement $\mathbf{z}\mathrm{d}x\mathrm{d}y\to \mathbf{x}\mathrm{d}y\mathrm{d}z$ and the integral plane selected to be any cross section $x=x_1$ of the DA.

Equation (9) can be understood intuitively. For Eqs. (9a) and (9c), the coefficient $a_{+}^{\text{sym/asym}}$ of the right-going SPP on the DA results from two contributions: the first one (${\alpha }_{+}^{\text{sym/asym}}$) is from the direct coupling of the up-going SPP on the waveguide; the second one is from the reflection ($r_b^{\text{sym/asym}}$) of the damped ($u^{\text{sym/asym}}$) left-going SPP (with coefficient $b_{+}^{\text{sym/asym}}$) at the left terminal of the DA coupled with the waveguide. Equations (9b) and (9d) can be understood in a similar way. Solving Eq. (9), one can obtain $$a_{+}=\frac{{\alpha }_{+}}{1-u^2{r}_a{r}_b},$$(11a)$$b_{+}=\frac{{\alpha }_{+}u{r}_a}{1-u^2{r}_a{r}_b},$$(11b)where the superscripts “sym” and “asym” are omitted since $a_{+}^{\mathrm{sym}}$ and $a_{+}^{\mathrm{asym}}$ ($b_{+}^{\mathrm{sym}}$ and $b_{+}^{\mathrm{asym}}$) have exactly the same form of expression. For the case of an incident backward/down-going SPP from the top of the waveguide, the coefficients of the excited right-going and left-going symmetric (antisymmetric) SPPs on the DA are denoted by $a_{-}^{\mathrm{sym}}$ and $b_{-}^{\mathrm{sym}}$ ($a_{-}^{\mathrm{asym}}$ and $b_{-}^{\mathrm{asym}}$), respectively, as shown in Fig. 3(a1). The coefficient of the right-going symmetric (antisymmetric) SPP on the semi-infinite DA excited by the down-going SPP on the waveguide is denoted by ${\alpha }_{-}^{\mathrm{sym}}$ (${\alpha }_{-}^{\mathrm{asym}}$), as shown in Fig. 3(a2). Fully parallel to Eq. (11), through building up an SPP model, one can obtain $$a_{-}=\frac{{\alpha }_{-}}{1-u^2{r}_a{r}_b},$$(12a)$$b_{-}=\frac{{\alpha }_{-}u{r}_a}{1-u^2{r}_a{r}_b},$$(12b)where the superscripts “sym” and “asym” are omitted as well.

The electromagnetic field in the coupling region between the SA and the right terminal of the DA excited by an incident up-going or down-going SPP on the waveguide can be respectively expressed as $${\mathit{\psi }}_{+}^{\text{coupling}}(\mathbf{r})=a_{+}^{\mathrm{sym}}{u}^{\mathrm{sym}}{\mathit{\psi }}_R^{\mathrm{sym}}(\mathbf{r})+a_{+}^{\mathrm{asym}}{u}^{\mathrm{asym}}{\mathit{\psi }}_R^{\mathrm{asym}}(\mathbf{r}),$$(13a)$${\mathit{\psi }}_{-}^{\text{coupling}}(\mathbf{r})=a_{-}^{\mathrm{sym}}{u}^{\mathrm{sym}}{\mathit{\psi }}_R^{\mathrm{sym}}(\mathbf{r})+a_{-}^{\mathrm{asym}}{u}^{\mathrm{asym}}{\mathit{\psi }}_R^{\mathrm{asym}}(\mathbf{r}),$$(13b)where ${\mathit{\psi }}_R^{\mathrm{sym}}$ (${\mathit{\psi }}_R^{\mathrm{asym}}$) represents the field in the coupling region excited by the right-going symmetric (antisymmetric) SPP on the DA with unitary coefficient [as shown in Fig. 3(a3)], and can be calculated with the fully vectorial a-FMM. To achieve the chiral coupling, a circularly polarized point source is placed in the gap between the DA and the SA, with coordinate ${\mathbf{r}}_0=(D/2+L+\mathrm{\Delta }x,0,0)$ and $\mathrm{\Delta }x$ to be determined hereafter [see the coordinate in Fig. 2(a)]. Substituting Eq. (13) into Eq. (3), one can obtain the coefficients of the up-going and down-going SPPs excited by the circularly polarized point source: $${\beta }_{+}=\frac{a_{-}^{\mathrm{sym}}{u}^{\mathrm{sym}}{E}_{x,R}^{\mathrm{sym}}({\mathbf{r}}_0)+i{a}_{-}^{\mathrm{asym}}{u}^{\mathrm{asym}}{E}_{z,R}^{\mathrm{asym}}({\mathbf{r}}_0)}{⟨{\mathit{\psi }}_{-}|{\mathit{\psi }}_{+}⟩},$$(14a)$${\beta }_{-}=\frac{a_{+}^{\mathrm{sym}}{u}^{\mathrm{sym}}{E}_{x,R}^{\mathrm{sym}}({\mathbf{r}}_0)+i{a}_{+}^{\mathrm{asym}}{u}^{\mathrm{asym}}{E}_{z,R}^{\mathrm{asym}}({\mathbf{r}}_0)}{⟨{\mathit{\psi }}_{-}|{\mathit{\psi }}_{+}⟩},$$(14b)where ${\mathit{\psi }}_{+}$ and ${\mathit{\psi }}_{-}$ represent the up-going and down-going fundamental SPP modes on the waveguide, respectively, $E_{x,R}^{\mathrm{sym}}$ is the $x$ component of the electric field of ${\mathit{\psi }}_R^{\mathrm{sym}}$, and $E_{z,R}^{\mathrm{asym}}$ is the $z$ component of the electric field of ${\mathit{\psi }}_R^{\mathrm{asym}}$. In Eq. (14), the symmetry of the structure with respect to the $z=0$ plane yields $$a_{+}^{\mathrm{sym}}=a_{-}^{\mathrm{sym}}=a^{\mathrm{sym}},a_{+}^{\mathrm{asym}}=-a_{-}^{\mathrm{asym}}=a^{\mathrm{asym}},$$(15)which is consistent with Eq. (4). Equation (14) indicates that the two SPPs with different symmetries on the DA provide two regulable coefficients ${\mathit{a}}^{\mathrm{sym}}{u}^{\mathrm{sym}}$ and ${\mathit{a}}^{\mathrm{a}\mathrm{sym}}{u}^{\mathrm{a}\mathrm{sym}}$ for the two orthogonal electric-field components $E_{x,R}^{\mathrm{sym}}({\mathbf{r}}_0)$ and $E_{z,R}^{\mathrm{asym}}({\mathbf{r}}_0)$ that constitute the T-spin, respectively. Specifically, Eqs. (11a) and (12a) show that ${\mathit{a}}^{\mathrm{sym}}{u}^{\mathrm{sym}}$ and ${\mathit{a}}^{\mathrm{a}\mathrm{sym}}{u}^{\mathrm{a}\mathrm{sym}}$ can be regulated by changing the length $L$ of the DA [via the phase shift factor $u^{\text{sym/asym}}=\exp (i{k}_0{n}_{\mathrm{eff}}^{\text{sym/asym}}L)$], which allows the possibility to form a T-spin field for an incident SPP, or reciprocally, the chiral coupling of the SPP, as to be shown in Fig. 4.

After obtaining ${\beta }_{\pm }$, one can then calculate ${\mathrm{\Gamma }}_{\mathrm{SPP},+}$ and ${\mathrm{\Gamma }}_{\mathrm{SPP},-}$, the coupling rates of the up-going and down-going SPPs on the waveguide excited by the point source, which are expressed as ${\mathrm{\Gamma }}_{\mathrm{SPP},\pm }=|{\beta }_{\pm }{|}^2{\iint }_{z=z_0}\frac{1}{2}\mathrm{Re}({\mathbf{E}}_{\pm }\times {\mathbf{H}}_{\pm }^{*})·(\pm \mathbf{z})\mathrm{d}x\mathrm{d}y$, with $z=z_0$ being the cross section of the waveguide where the SPP begins to propagate. The curves of ${\mathrm{\Gamma }}_{\mathrm{SPP},\pm }$ as functions of the length $L$ of the DA are shown in Fig. 4(a1), with ${\mathrm{\Gamma }}_{\mathrm{SPP},\pm }$ normalized by ${\mathrm{\Gamma }}_{\mathrm{air}}$, the spontaneous emission rate of the circularly polarized point source ${\mathbf{j}}_{\sigma }$ in air. ${\mathrm{\Gamma }}_{\mathrm{air}}={\eta }_{\mathrm{vac}}{k}_0^2{n}_a/(6\pi )$ with ${\eta }_{\mathrm{vac}}$ being the wave impedance in vacuum, and $n_a=1$ being the refractive index of air. In the calculation, the source is located at ${\mathbf{r}}_0=(D/2+L+7\mathrm{nm},0,0)$. Note that a localization accuracy of a single quantum-dot emitter better than 2 nm has been achieved experimentally [80]. In Fig. 4(a), the results obtained with the fully vectorial a-FMM and SPP model [Eq. (14)] are shown with circles and solid curves, respectively. Good agreement can be observed between the two results, which confirms the validity of the model. Here the calculation of the ${\beta }_{\pm }$ with the a-FMM is the same as Eq. (10): the total field excited by the source is calculated with the a-FMM first, and then the ${\beta }_{\pm }$ are extracted from the total field with the mode orthogonality theorem [73,79].

Figure 4(a1) shows that for some specific values of $L$ (at the green vertical dashed lines of $n=0$, 1, 3 in the figure), one of ${\mathrm{\Gamma }}_{\mathrm{SPP},+}$ and ${\mathrm{\Gamma }}_{\mathrm{SPP},-}$ reaches the peak value and obtains significant enhancement (${\mathrm{\Gamma }}_{\mathrm{SPP},\pm }/{\mathrm{\Gamma }}_{\mathrm{air}}\gg 1$), while the other is almost zero. This means the occurrence of perfect chiral coupling of the SPP waveguide mode, and the corresponding $L$ are called perfect chiral points (PCPs) here. In addition, without changing the chirality of the source, the unidirectional excitation of the SPP on the waveguide can be reversed in direction (between $n=0$, 1 and $n=3$) or disappear (at $n=2$) simply by changing $L$. The $L$ corresponding to the latter are called perfect non-chiral points (PNCPs). At the PCPs of $n=0$, 1, 3, the footprints of the coupling structure are $L+G+D=102$, 245, 560 nm in the $x$ direction, respectively, and are all $L_{\mathrm{SA}}=370\mathrm{nm}$ in the $z$ direction, which are much smaller than the footprints (from several to more than 100 μm) of the dielectric whispering-gallery microcavities with chiral-coupling structures [32,34,50,63].

To explain the above numerical phenomena, Figs. 4(a2) and 4(a3) respectively show the moduli of the coefficients of the antisymmetric and symmetric SPPs on the DA, $|a^{\mathrm{asym}}|$ and $|a^{\mathrm{sym}}|$, which are given by model Eq. (11a) or (12a). The results reveal that the PCPs and PNCPs shown in Fig. 4(a1) almost exactly correspond to the different orders of FP resonance of the antisymmetric SPP on the DA, which are achieved under the phase-matching condition, $$2k_0\mathrm{Re}(n_{\mathrm{neff}}^{\mathrm{asym}})L+\arg (r_a^{\mathrm{asym}})+\arg (r_b^{\mathrm{asym}})=2n\pi .$$(16)Equation (16) is obtained by minimizing the denominator in Eq. (11a) or (12a) for $a^{\mathrm{asym}}$, and resultantly, maximizing $|a^{\mathrm{asym}}|$ [see Fig. 4(a2)]. In Eq. (16), Re(·) denotes the real part, arg(·) denotes the argument, and $n$ is an integer corresponding to different orders of resonance [shown by the green vertical dashed lines in Figs. 4(a)–4(c)]. Equation (16) can be understood intuitively as follows: the phase change accumulated by the antisymmetric SPP that propagates back and forth over one round on the DA is multiples of $2\pi$, which obviously results in a constructive interference of the multiple-scattered SPPs and thus forms the FP resonance.

To analyze the impact of the FP resonance on the chirality of the electric field at the position of the source, Eq. (13) gives that the two orthogonal components of the electric field excited by the incident SPP on the waveguide are $$E_{x,+}^{\text{coupling}}({\mathbf{r}}_0)=E_{x,-}^{\text{coupling}}({\mathbf{r}}_0)=E_x^{\text{coupling}}({\mathbf{r}}_0)=a^{\mathrm{sym}}{u}^{\mathrm{sym}}{E}_{x,R}^{\mathrm{sym}}({\mathbf{r}}_0),$$(17a)$$E_{z,+}^{\text{coupling}}({\mathbf{r}}_0)=-E_{z,-}^{\text{coupling}}({\mathbf{r}}_0)=E_z^{\text{coupling}}({\mathbf{r}}_0)=a^{\mathrm{asym}}{u}^{\mathrm{asym}}{E}_{z,R}^{\mathrm{asym}}({\mathbf{r}}_0).$$(17b)For the symmetric SPP on the DA, Fig. 4(a3) shows that the value of $|a^{\mathrm{sym}}|$ is smaller with a weak fluctuation compared to $|a^{\mathrm{asym}}|$ at resonance as the length $L$ of the DA varies, which implies a weaker FP resonance of the symmetric SPP than that of the antisymmetric SPP. However, since the SA at a specific FP resonance is placed near the right terminal of the DA, the value of $|E_{x,R}^{\mathrm{sym}}({\mathbf{r}}_0)|$ is larger than that of $|E_{z,R}^{\mathrm{asym}}({\mathbf{r}}_0)|$ [$E_{x,R}^{\mathrm{sym}}({\mathbf{r}}_0)=3.9838–11.358i$ and $E_{z,R}^{\mathrm{asym}}({\mathbf{r}}_0)=1.3206–5.3317i$], which thus compensates for the smaller value of $|a^{\mathrm{sym}}|$ than that of $|a^{\mathrm{asym}}|$ at resonance, so that the relation $|E_x^{\text{coupling}}({\mathbf{r}}_0)|\approx |E_z^{\text{coupling}}({\mathbf{r}}_0)|$ can be achieved under the phase-matching condition of Eq. (16). The above analysis confirms the expectation of using the DA and SA to respectively enhance the $E_z$ and $E_x$ components at the source position in the gap between the DA and the SA as stated in Section 3.A.

Next, we check the phase difference $\phi =\arg [E_z^{\text{coupling}}({\mathbf{r}}_0)/E_x^{\text{coupling}}({\mathbf{r}}_0)]$ of the two orthogonal components of the electric field at the position of the source under the phase-matching condition. Figure 4(b) shows the $\phi$ as a function of $L$ for an incident up-going SPP on the waveguide. Corresponding to $n=0$, 1, 2, 3 in Eq. (16), there are $\phi \approx \pi /2$, $\pi /2$, 0, $-\pi /2$, i.e., the right-handed, right-handed, null, and left-handed T-spins of the electric field are excited at ${\mathbf{r}}_0$, respectively. Accordingly, for the original problem, if a right-handed circularly polarized source is placed at ${\mathbf{r}}_0$ with $L$ taking values corresponding to $n=0$, 1, 2, 3 in Eq. (16), the forward, forward, bidirectional, and backward SPP will be launched on the waveguide, respectively.

To quantitatively describe the directional excitation of the up-going or down-going SPP on the waveguide by the chiral source, Fig. 4(c) shows the dependence of the directivity factor $g$ on $L$. For the $L$ given by Eq. (16) with $n=0$, 1, 2, 3, there are $g=0.842$, 0.999, 0.0510, $-0.892$, respectively. In addition, $g$ can take values continuously over almost the whole range of [−1, 1] simply by changing $L$ (instead of changing the chirality of the source [29,37,44–46,48] or putting the source on the other side of the waveguide [57,59]), which provides flexible freedoms for the design of chiral-coupling devices. Note that the poorer negative value of $g=-0.892$ at $n=3$ than the positive value of $g=0.999$ at $n=1$ can be further improved by adjusting the excitation and structural parameters with the aid of Eq. (7) and the SPP model. For instance, if the point source is moved from ${\mathbf{r}}_0=(D/2+L+7\mathrm{nm},0,0)$ to ${\mathbf{r}}_0=(D/2+L+5\mathrm{nm},0,0)$, $g$ at $n=3$ can be improved to be −0.964, with a moderate decrease of $g$ at $n=1$ to 0.959.

Next we will provide a direct numerical observation for the formation of the chiral/non-chiral light field in the reciprocal problem, and for the unidirectional/bidirectional excitation of the SPP on the waveguide in the original problem, as predicted by the SPP model above. For the reciprocal problem, we calculate the spatial distribution of the Stokes parameter $S_3$ defined before Eq. (7) under the excitation of an up-going SPP on the waveguide. The results are obtained with the model [Eq. (13a)] and are shown in Fig. 4(d). It can be seen that for $L=185$, 342, 500 nm (corresponding to $n=1$, 2, 3), right-handed circularly polarized, linearly polarized, and left-handed circularly polarized electric fields at ${\mathbf{r}}_0$ [corresponding to $S_3({\mathbf{r}}_0)=0.999$, 0.0151, −0.897] are excited by the up-going SPP on the waveguide, respectively. For the original problem, a right-handed circularly polarized source is placed at ${\mathbf{r}}_0$ and $L$ also takes the above three values. The distributions of the excited electric field are calculated with the finite element method (FEM) executed with the software of COMSOL Multiphysics. As shown in Fig. 4(e), the results manifest that for these three values of $L$, forward, bidirectional, and backward SPPs (corresponding to $g=0.999$, 0.0510, −0.892 obtained with the a-FMM) are excited on the waveguide, respectively. These results confirm the prediction of the SPP model and also the validity of the general relation Eq. (7) [note that the slight difference between $S_3({\mathbf{r}}_0)$ and $g$ is due to the slight error of the SPP model relative to the a-FMM calculation].

In addition, Fig. 4(e) shows that the SA is excited to the designed FP resonance [see Fig. 9(b) in Appendix B.1], which enhances the electric-field component $E_x$ in the gap between the DA and the SA, and thus meets the requirement for a chiral coupling as described in a previous subsection. Meanwhile, one can observe a standing wave formed by the two counter-propagating antisymmetric SPPs on the DA at FP resonance, and the number of nodes of field increases with the increase of $L$.

The general relation of Eq. (8) predicts that if the coupling structure is designed to satisfy $S_3({\mathbf{r}}_0)=1$ (i.e., to achieve a perfect chiral coupling), the polarization of the excitation source ($S_{3,\mathrm{source}}$) will be directly related to the directivity of the excited SPP ($g$), thus demonstrating the feasibility to read out the static qubit of the source ($S_{3,\mathrm{source}}$) by measuring the flying qubit of photons ($g$) [1,46]. To confirm this prediction, we fix the structure with $L=185\mathrm{nm}$ [corresponding to $n=1$ in Eq. (16)], for which there is $S_3({\mathbf{r}}_0)=0.999$, and check the excitation of the SPP waveguide mode by a point source of various polarizations. Figure 4(e) with $n=1$ and Fig. 5(a) show that the direction of the excited SPP will be reversed if the source is changed from right-handed to left-handed circular polarization. Figures 5(b)–5(d) show that the excitation of the SPP becomes bidirectional if the source is of linear or elliptical polarization. More specifically, there are ${\mathrm{\Gamma }}_{\mathrm{SPP},+}/{\mathrm{\Gamma }}_{\mathrm{air}}({\mathrm{\Gamma }}_{\mathrm{SPP},-}/{\mathrm{\Gamma }}_{\mathrm{air}})=40.36(0.01268),0.01268(40.36),20.46(20.46),30.51(10.33),10.33(30.51)$, and $g=0.999$, −0.999, 0.00768, 0.494, −0.494 (obtained with the a-FMM) for Fig. 4(e) with $n=1$ and Figs. 5(a)–5(d), for which $S_{3,\mathrm{source}}=1$, −1, 0, 0.5, −0.5, respectively. Therefore, the prediction of the general relation Eq. (8) is confirmed. Note that the slight difference between $g$ and $S_{3,\mathrm{source}}$ is due to the slight error of $S_3({\mathbf{r}}_0)$ relative to 1.

The enhancement of the spontaneous emission rate is described by $F_P={\mathrm{\Gamma }}_{\mathrm{total}}/{\mathrm{\Gamma }}_{\mathrm{air}}$, called the Purcell factor [81,82], where ${\mathrm{\Gamma }}_{\mathrm{total}}$ is the total spontaneous emission rate (proportional to emission power) of the source in the designed structure. A high value of $F_P$ is important for the realization of an on-chip ultrafast emission source [83]. For the chiral-coupling system studied in this paper, higher $F_P$ means a faster readout of spin qubits from the source, that is, a higher conversion rate from the static qubits of matter to the flying qubits of photons [66]. For a right-handed circularly polarized electric dipole source ${\mathbf{j}}_{\sigma }=(\mathbf{x}+\mathbf{z}i)\delta (\mathbf{r}-{\mathbf{r}}_0)$, there is ${\mathrm{\Gamma }}_{\mathrm{total}}=-\{\mathrm{Re}[E_x({\mathbf{r}}_0)]+\mathrm{Im}[E_z({\mathbf{r}}_0)]\}/2$, where $E_x({\mathbf{r}}_0)$ and $E_z({\mathbf{r}}_0)$ are respectively the $x$ and $z$ components of the electric field excited by the source at ${\mathbf{r}}_0$.

The coupling efficiency of the SPP on the waveguide excited by the source is defined as ${\eta }_{\pm }={\mathrm{\Gamma }}_{\mathrm{SPP},\pm }/{\mathrm{\Gamma }}_{\mathrm{total}}$ [1,64,66,81], which describes the probability for emitted photons to be channeled into the up-going or down-going SPP waveguide mode [66]. Improvement of ${\eta }_{\pm }$ is of great significance for improving the signal intensity of the waveguide mode in the on-chip single photon source and integrated photonic circuits [1,37].

To analyze the $F_P$ and ${\eta }_{\pm }$, an SPP model can be built up as well for the original problem [as illustrated in Fig. 3(b) and detailed in Appendix B.2]. With the model [Eq. (B3) in Appendix B.2], the total spontaneous emission rate can be expressed as $${\mathrm{\Gamma }}_{\text{total}}=-\frac{1}{2}\mathrm{Re}[E_{x,\text{source}}({\mathbf{r}}_0)+a_s^{\mathrm{sym}}{u}^{\mathrm{sym}}{E}_{x,R}^{\mathrm{sym}}({\mathbf{r}}_0)]-\frac{1}{2}\mathrm{Im}[E_{z,\text{source}}({\mathbf{r}}_0)+a_s^{\mathrm{asym}}{u}^{\mathrm{asym}}{E}_{z,R}^{\mathrm{asym}}({\mathbf{r}}_0)].$$(18)Equation (18) shows that the two SPPs with different symmetries on the DA provide two regulable coefficients $a_s^{\mathrm{sym}}{u}^{\mathrm{sym}}$ and $a_s^{\mathrm{asym}}{u}^{\mathrm{asym}}$ for ${\mathrm{\Gamma }}_{\mathrm{total}}$, which depend on the length $L$ of the DA [via the phase shift factor $u$; see Eq. (B2a) in Appendix B.2].

In addition, with the SPP model for the original problem, analytical expression of the coefficient ${\beta }_{\pm }$ of the SPP waveguide mode excited by the source can be obtained as well; see Eq. (B4) in Appendix B.2. As expected, with the reciprocity theorem [72,73] it can be proved that the ${\beta }_{\pm }$ given by the SPP models for the original problem and the reciprocal problem [see Eq. (14)] are identical, as shown in Appendix B.2.

The dependence of the Purcell factor $F_P$ on the length $L$ of the DA is shown in Fig. 6(a). The results obtained with the SPP model [Eq. (18)] are in good agreement with the rigorous numerical results of the a-FMM, which confirms the validity of the model. At the PCPs and PNCPs ($n=0$, 1, 2, 3) shown in Fig. 4(a1), $F_P$ reaches high peak values of 119, 203, 152, and 145, respectively, exhibiting a significant enhancement of the spontaneous emission rate of the chiral source. These high peak values of $F_P$ result from the fact that under the phase-matching condition of Eq. (16), the coefficient $a_s^{\mathrm{asym}}$ of the antisymmetric SPP in Eq. (18) will take a maximum value.

The results of the coupling efficiency ${\eta }_{\pm }$ of the SPP on the waveguide are shown in Fig. 6(b). At the PCPs and PNCPs ($n=0$, 1, 2, 3), there are ${\eta }_{+}=44\%$ (${\eta }_{-}=4.7\%$), 25% (0.028%), 10% (10%), and 0.87% (16%), respectively, exhibiting an efficient unidirectional or bidirectional coupling of the SPP. The values of ${\eta }_{\pm }$ could be further improved by optimizing the system parameters with the aid of the SPP model, for instance, to suppress the SPP propagation loss [84] or non-radiative decay rate [81,85] by optimizing the sizes and materials of the DA and the SA and their in-between gaps.

As shown in Appendix B.3, the system without the SA exhibits a weaker unidirectionality of the excited SPP waveguide mode [Figs. 10(a1) and 10(b1)]. For a bare SPP waveguide (acting as a direct chiral-coupling system [57,59]) in comparison with the proposed indirect chiral-coupling system, the former exhibits obviously poorer performances, regarding the effective-coupling distance, the directivity factor $g$, the coupling efficiency ${\eta }_{\pm }$ of the SPP waveguide mode, and the Purcell factor $F_P$ of the chiral source [Figs. 10(a2), 10(b2), and 11]. These comparative results show the performance advantages of the proposed indirect chiral-coupling system.

As shown in Appendix B.4 (Fig. 12), the performances of the proposed indirect chiral-coupling system can be largely preserved after adding a substrate and considering the actual location of a quantum-dot emitter on the substrate [37,39,44–46,48,64]. This exhibits the robustness of performances and feasibility of an experimental demonstration for the proposed indirect chiral-coupling system with practical configurations.

We propose a general approach based on the reciprocity theorem for an intuitive analysis and rigorous calculation of the coupling coefficient between the chiral source and the waveguide mode. With this approach, we derive the conditions for the occurrence of chiral coupling as well as some general relations between the chiral-coupling directivity ($g$) and T-spin ($S_3$) of the field or source.

Based on the theories, a novel indirect chiral-coupling system between a chiral emitter and waveguide modes mediated by FP resonance is proposed, which is on the platform of SPP with a deep subwavelength scale. Compared with the direct chiral-coupling system, this system gets rid of the restriction that the emission source must be in the limited evanescent-field region of the waveguide mode. As a mediator, the FP resonance provides flexible freedoms to regulate the chiral coupling, and can achieve nearly perfect chiral coupling, non-chiral coupling, and a direction reversal of the chiral coupling without changing the chirality of the source. In addition, with the assistance of the FP resonance, high spontaneous-emission-enhancement Purcell factor of the chiral source and high chiral-coupling efficiency between the source and the SPP waveguide mode are obtained, which enables the system to realize a deterministic, fast, and efficient readout of the spin qubits in the source. The proposed chiral-coupling system is expected to be realizable in experiment in view of the current fabrication and testing capabilities [77,80] and our simulation results for practical configurations.

To explore the underlying physical mechanism of the indirect chiral-coupling system, two SPP models based on first principles are built up by considering the excitation and multiple scattering processes of SPPs in the structure. The SPP model for the reciprocal problem indicates that the T-spin at the position of the source originates from two SPPs with different symmetries supported by the structure, and that an emergence, disappearance, and flip of T-spin can be realized simply by adjusting the antisymmetric SPP to reach different orders of FP resonance. The SPP model for the original problem clarifies that once the antisymmetric SPP is at an FP resonance, the spontaneous-emission-enhancement Purcell factor will reach the maximum, and the chiral-coupling efficiency between the source and the SPP waveguide mode will take a large value.

We expect that our proposed theories will provide general recipes for an intuitive and quantitative design of various direct/indirect chiral-coupling systems. Thanks to the FP resonance with flexible design freedoms and rich implementation platforms (such as SPP platforms [79,86] and photonic-mode dielectric platforms [87,88]), indirect chiral-coupling systems with the FP resonance as coupling intermediaries can be further developed based on the work of this paper to achieve improved performances and extended applications.