Controllable intermodal coupling in waveguide systems based on tunable hyperbolic metamaterials

Anna Tyszka-Zawadzka; Bartosz Janaszek; Marcin Kieliszczyk; Paweł Szczepański; Paweł Szczepański

doi:10.1364/OE.413825

1. Introduction

Intermodal coupling and power transfer is a working principle of many optical guided-wave devices such as waveguide couplers [1–22], laser arrays [23,24], distributed feedback lasers [25,26], filters [27], modulators, and switches [1,28]. In particular, the role of intermodal coupling is of key importance in optical integrated systems, providing means for lossless power exchange between planar waveguides [1–11,15–17].

More recently, a class of novel anisotropic media exhibiting hyperbolic dispersion, known as hyperbolic metamaterials (HMMs), has been recognized as a very prospective building block for planar waveguide applications. Until now, it has been demonstrated that waveguides based on HMMs reveal a number of unusual properties, such as simultaneous propagation of plasmons and bulk waves [29,30], field enhancement [31], coexistence of two modes with the same direction of phase velocity and the contrary signs of energy flows (known as forward and backward modes), and light slowing or stopping [32,33]. More recently, a concept of controlling properties of metamaterial structures by adjusting their geometry has been gathering widespread attention [34,35]. In particular, a possibility of active tuning optical properties of hyperbolic metamaterials by incorporating different stimulus-sensitive materials has been intensively investigated [36–43]. In particular, a great deal of attention has been devoted to waveguides based on tunable hyperbolic metamaterials (THMMs) [44–47] revealing controlled plasmon propagation [44], controlled light slowing or stopping in terahertz [45] as well as near-infrared spectral range [46] and ultra-far mode propagation [47]. These unique waveguiding properties constitute THMM-based waveguides as a prospective constituent for novel integrated systems, particularly directional couplers of new functionalities. So far, the studies concerning waveguiding properties of HMM/THMM structure have been limited to single-waveguide systems. What is more, until now, intermodal coupling in any waveguide system based on HMM/THMM has not been yet investigated.

In literature, there exist a number of methods allowing coupled-mode analysis in various waveguiding systems. One of the first reported approaches, known as the conventional coupled-mode theory, is based on the perturbation technique and the assumption of the orthogonality of waveguide modes [1,2]. Further developments in this field were focused on embracing the effect of nonorthogonality between waveguide modes. In particular, the modified mode-coupling theory [6], as well as formalisms based on the variational principle [7,8,13], or the Lorentz reciprocity theorem [13–15,19], have been formulated. Especially, an approach exploiting the Lorentz reciprocity theorem is particularly interesting in terms of its versatility for various systems, such as multi-waveguide and/or multi-mode structures [14,15], anisotropic media [15] and nonlinear couplers [19]. Moreover, this approach is indispensable to correctly describe the coupling between forward- and backward-propagating modes in anisotropic and bianisotropic reciprocal waveguides with gain and loss [48].

Within this paper, we present a rigorous analytical approach, based on the Lorentz reciprocity theorem, enabling analysis of modal coupling in the waveguiding system incorporating a tunable hyperbolic metamaterial based on graphene (further regarded as THMM), which has not been yet investigated. The considered system consists of two parallel planar waveguides with core layers formed by an isotropic dielectric material and graphene-based anisotropic medium revealing Type II hyperbolic dispersion [49,50]. In the course of this analysis we investigate influence of parameters of the considered system, such as waveguide width or separation distance between waveguides, on the strength of intermodal coupling. Moreover, we demonstrate that due to the presence of graphene, it is possible to control optical properties of the THMM medium, and consequently, the power exchange within the considered system, via change of chemical potential of graphene.

The manuscript is structured as follows: in Section 2, we present a theoretical model of modal coupling in the considered waveguiding system by using the fundamental relations of the generalized reciprocity theorem [15]. Next, we analyze a number of characteristics revealing the effect of geometrical structure parameters and the Fermi energy (for example controlled by external electric voltage) on the coupling coefficients and the power transfer between the TM mode of the dielectric waveguide and the forward- and backward modes of the THMM waveguide. For clarity of our analysis, all the presented data are supported with relations between transverse and longitudinal field components (see Appendix A1), as well as the exact analytical expressions for the coefficient of the coupled mode equations describing the representative system (see Appendix A2).

2. Theoretical model

In this section, we describe the coupled-waveguide system considered, see Figs. 1(a)–1(c), and derive corresponding coupled-mode equations (based on Lorentz reciprocity theorem) to analyze intermodal coupling controlled by chemical potential of graphene.

Fig. 1. Scheme of the standalone dielectric waveguide (a), the THMM waveguide (b) and the complete coupled-waveguide system (c).

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2.1 Considered system

The considered system consists of two parallel planar waveguides with core layers formed by an isotropic dielectric, described with electric permittivity ɛ_a, and a uniaxial anisotropic THMM medium, described with permittivity tensor $\overline{\bar{\varepsilon}_{b}}=\operatorname{diag}\left(\left[\begin{array}{lll}\varepsilon_{\|}, \varepsilon_{\|}, \varepsilon_{\perp}\end{array}\right]\right)$ with its components obtained with the help of the effective medium theory (EMT) [30,49]. The core layers are symmetrically cladded with isotropic dielectric material characterized by relative electric permittivity ɛ_c, see Figs. 1(a)–1(c).

Without loss of generality, we assume that the dielectric waveguide supports only fundamental modes, i.e., TM₀ and TE₀. Moreover, the THMM core layer, consisting of periodically arranged graphene and dielectric bilayers, is assumed to effectively reveal Type II hyperbolic dispersion, which provides simultaneous existence of forward and backward-propagating TM modes having the same order and phase velocity direction, but antiparallel energy flux [46]. It is worth noting that in such a waveguide, in contrast to TM modes, the propagation of TE modes is strongly suppressed [51]. Moreover, the coupling between TE₀ mode of the dielectric waveguide and TM modes of the THMM waveguide is not possible. Thus, in further analysis we focus only on the coupling between TM modes in the considered system.

Furthermore, since graphene’s permittivity can be altered with chemical potential [42], the propagation properties of the considered THMM waveguide, and consequently of the complete coupled-waveguide system, may be modified in the same manner, revealing novel functionality arising from unique dispersion properties of the THMM medium.

2.2 Coupled-mode equations

We start from the Lorentz reciprocity theorem for anisotropic media [15], which will further serve as a theoretical basis for the formulation of coupled-mode equations for the given system.

Consider two sets of field solutions (E⁽¹⁾, H⁽¹⁾) and (E⁽²⁾, H⁽²⁾), which satisfy the two Maxwell’s equations in a nonmagnetic media described by relative electric permittivity tensors $\overline{\bar{\varepsilon}}^{(1)}(x, y)$ and $\overline{\bar{\varepsilon}}^{(2)}(x, y)$ and relative scalar magnetic permittivity µ⁽¹⁾= µ⁽²⁾ = 1, i.e.,

(1a)$$\nabla \times {{\boldsymbol E}^{(1,2)}} = i\omega {\mu ^{(1,2)}}{{\boldsymbol H}^{(1,2)}},$$

(1b)$$\nabla \times {\boldsymbol H}^{(1,2)} = -i\omega \overline{\bar{\varepsilon }}^{(1,2)}{\boldsymbol E}^{(1,2)},$$

with appropriate boundary conditions. By combining Eqs. (1a) and (1b), it is possible to obtain the following relation between the two sets of fields:

(2)$$\nabla \cdot ({{{\boldsymbol E}^{(1)}} \times {{\boldsymbol H}^{(2)}} - {{\boldsymbol E}^{(2)}} \times {{\boldsymbol H}^{(1)}}} )= i\omega {{\boldsymbol E}^{(1)}} \cdot ({{{\overline{\bar{\varepsilon }}}^{(2)}}(x,y) - {{\overline{\bar{\varepsilon }}}^{(1)}}(x,y)} ){{\boldsymbol E}^{(2)}},$$

corresponding to the Lorentz reciprocity theorem. Here, the time convention exp(-iωt) is used, where ω is the frequency of light. Applying the above relation to an infinitesimal section Δz of a geometry, which is translational invariant in the z direction, the following identity can be obtained:

(3)$$\begin{array}{l} \frac{\partial }{{\partial z}}\int {\int {({{{\boldsymbol E}^{(1)}} \times {{\boldsymbol H}^{(1)}} - {{\boldsymbol E}^{(2)}} \times {{\boldsymbol H}^{(2)}}} )\cdot \hat{z}dxdy} } \\ = i\omega \int {\int {({{{\overline{\bar{\varepsilon }}}^{(2)}}(x,y) - {{\overline{\bar{\varepsilon }}}^{(1)}}(x,y)} ){{\boldsymbol E}^{(1)}} \cdot {{\boldsymbol E}^{(2)}}dxdy} } , \end{array}$$

which is known as the generalized reciprocity relation [15]. It is worth noting, that the above relation is exact and only requires fields, which satisfy the Maxwell’s equations with appropriate boundary conditions. We use this equation to derive a set of coupled-mode equations for the directional coupler shown in Fig. 1(c).

2.2.1 Step 1

As the first system (further regarded as system <1>) with the field solutions (E⁽¹⁾, H⁽¹⁾), we choose the complete coupler characterized by the relative permittivity of the following spatial distribution:

(4)$${\overline{\bar{\varepsilon }}^{(1)}} = \overline{\bar{\varepsilon }}(x) = \left[ {\begin{array}{ccc} {{\varepsilon_{xx}}}&0&0\\ 0&{{\varepsilon_{yy}}}&0\\ 0&0&{{\varepsilon_{zz}}} \end{array}} \right] = \left\{ {\begin{array}{lc} {{\varepsilon_c} \cdot \overline{\bar{I}},}&{x < - a/2}\\ {{\varepsilon_a} \cdot \overline{\bar{I}},}&{ - a/2 < x < a/2}\\ {{\varepsilon_c} \cdot \overline{\bar{I}},}&{a/2 < x < d}\\ {{{\overline{\bar{\varepsilon }}}_b},}&{a/2 + d + b < x < a/2 + d}\\ {{\varepsilon_c} \cdot \overline{\bar{I}},}&{x > a/2 + d + b} \end{array},} \right.$$

where $\overline{\overline I} $ is a unit tensor, d is the distance between waveguides, while a and b are widths of the dielectric and THMM core layers, respectively. In our case, the set of fields (E⁽¹⁾, H⁽¹⁾) is the superposition of the fundamental TM₀ mode of the dielectric waveguide (further referred as TM_0(a) with $\left( {\left( {\begin{array}{ccc} {E_x^{(a) + },}&{0,}&{E_z^{(a) + }} \end{array}} \right),\left( {\begin{array}{ccc} {0,}&{H_y^{(a) + },}&0 \end{array}} \right)} \right)$), the fundamental forward TM₀ mode (further referred as TM_0(bF) with $\left( {\left( {\begin{array}{ccc} {E_x^{(bF) + },}&{0,}&{E_z^{(bF) + }} \end{array}} \right),\left( {\begin{array}{ccc} {0,}&{H_y^{(bF) + },}&0 \end{array}} \right)} \right)$) and the fundamental backward TM₀ mode (further referred as TM_0(bB) with $\left( {\left( {\begin{array}{ccc} {E_x^{(bB) + },}&{0,}&{E_z^{(bB) + }} \end{array}} \right),\left( {\begin{array}{ccccc} {0,}&{H_y^{(bB) + },}&0 \end{array}} \right)} \right)$) of the THMM waveguide, all propagating in the + z direction. It is worth noting, that in the chosen set of fields (E⁽¹⁾, H⁽¹⁾) higher order TM modes of the THMM waveguide are omitted. This assumption is justified since, a THMM waveguide can be designed in a way allowing simultaneous support of fundamental forward mode TM_0(bF) and infinite number of backward modes TM_n(bB). Moreover, the coupling to higher order TM_n(bB) modes may be neglected, due to high propagation-constant mismatch between the fundamental modes TM_0(a), TM_0(bF), TM_0(bB) and higher order modes TM_n(bB) (see Results and Discussion for more detailed discussion). Thus, the transverse field components of the set of fields (E⁽¹⁾, H⁽¹⁾) may be expressed as follows

(5a)$${\boldsymbol E}_t^{(1)} = E_x^{(1)} \cdot \hat{x} = ({a(z)E_x^{(a) + }(x) + {b_F}(z)E_x^{(bF) + }(x) + {b_B}(z)E_x^{(bB) + }(x)} )\cdot \hat{x},$$

(5b)$${\boldsymbol H}_t^{(1)} = H_y^{(1)} \cdot \hat{y} = ({a(z)H_y^{(a) + }(x) + {b_F}(z)H_y^{(bF) + }(x) + {b_B}(z)H_y^{(bB) + }(x)} )\cdot \hat{y}.$$

With the help of Maxwell’s equations, the longitudinal components of the fields may be obtained from the transverse components (see Appendix A1) and take the following form:

(6a)$${\boldsymbol E}_z^{(1)} = E_z^{(1)} \cdot \hat{z} = \left( {\frac{{a(z)}}{{{\varepsilon_{zz}}}}\varepsilon_{zz}^{(a)}E_z^{(a) + }(x) + \frac{{{b_F}(z)}}{{{\varepsilon_{zz}}}}\varepsilon_{zz}^{(b)}E_z^{(bF) + }(x) + \frac{{{b_B}(z)}}{{{\varepsilon_{zz}}}}\varepsilon_{zz}^{(b)}E_z^{(bB) + }(x)} \right) \cdot \hat{z},$$

(6b)$${\boldsymbol H}_z^{(1)} = H_z^{(1)} \cdot \hat{z} = 0.$$

where a(z), b_F(z), b_B(z) are mode amplitude of TM_0(a), TM_0(bF) and TM_0(bB), respectively, and $\varepsilon _{zz}^{(a)}$ is the z- component of the tensor ${\overline{\bar{\varepsilon }}^{(a)}}$ describing spatial distribution of relative permittivity of the standalone dielectric waveguide [see Fig. 1(a)] given by:

(7)$${\overline{\bar{\varepsilon }}^{(a)}}(x) = \left[ {\begin{array}{ccc} {\varepsilon_{xx}^{(a)}}&0&0\\ 0&{\varepsilon_{yy}^{(a)}}&0\\ 0&0&{\varepsilon_{zz}^{(a)}} \end{array}} \right] = \left\{ {\begin{array}{ccc} {{\varepsilon_c} \cdot \overline{\bar{I}},}&{x < - a/2}&{}\\ {{\varepsilon_a} \cdot \overline{\bar{I}},}&{ - a/2 < x < a/2}&{}\\ {{\varepsilon_c} \cdot \overline{\bar{I}},}&{x > a/2}&{} \end{array}} \right.,$$

and $\varepsilon _{zz}^{(b)}$ is the z- component of the tensor ${\overline{\bar{\varepsilon }}^{(b)}}$describing spatial distribution of permittivity of the standalone THMM waveguide [see Fig. 1(b)] given by:

(8)$${\overline{\bar{\varepsilon }}^{(b)}}(x) = \left[ {\begin{array}{ccc} {\varepsilon_{xx}^{(b)}}&0&0\\ 0&{\varepsilon_{yy}^{(b)}}&0\\ 0&0&{\varepsilon_{zz}^{(b)}} \end{array}} \right] = \left\{ {\begin{array}{ccc} {{\varepsilon_c} \cdot \overline{\bar{I}},}&{}&{a/2 + d < x}\\ {{{\overline{\bar{\varepsilon }}}_b} \cdot \overline{\bar{I}},}&{}&{a/2 + d + b < x < a/2 + d}\\ {{\varepsilon_c} \cdot \overline{\bar{I}},}&{}&{x > a/2 + d + b} \end{array}} \right..$$

As the second system (further regarded as system <2>) with the field solutions (E⁽²⁾, H⁽²⁾), we consider the standalone dielectric waveguide, i.e., ${\overline{\bar{\varepsilon }}^{(2)}}(x) = {\overline{\bar{\varepsilon }}^{(a)}}(x)$which is defined by Eq. (7) . Furthermore, we choose the field solutions (E⁽²⁾, H⁽²⁾) to be an fundamental oscillatory mode TM_0(a) with $\left( {\left( {\begin{array}{ccc} {E_x^{(a) - },}&{0,}&{E_z^{(a) - }} \end{array}} \right),\left( {\begin{array}{ccc} {0,}&{H_y^{(a) - },}&0 \end{array}} \right)} \right)$ field components propagating in the -z direction, i.e.:

(9a)$${{\boldsymbol E}^{(2)}} = ({{\boldsymbol E}_t^{(a) - } + {\boldsymbol E}_z^{(a) - }} ){e^{ - i{\beta _a}z}} = ({E_x^{(a) - } \cdot \hat{x} + E_z^{(a) - } \cdot \hat{z}} ){e^{ - i{\beta _a}z}},$$

(9b)$${{\boldsymbol H}^{(2)}} = ({{\boldsymbol H}_t^{(a) - } + {\boldsymbol H}_z^{(a) - }} ){e^{ - i{\beta _a}z}} = ({H_y^{(a) - } \cdot \hat{y}} ){e^{ - i{\beta _a}z}}.$$

where ${\beta _a}$ is the propagation constant of the TM_0(a) mode.

Substituting (4)-(9) into both sides of the general reciprocity relation (3), we obtain the following coupled mode equation:

(10)$$\begin{array}{c} {{C_{(a)(a)}}\frac{{da}}{{dz}} + C_{(a)(bF)}^{}\frac{{d{b_F}}}{{dz}} + C_{(a)(bB)}^{}\frac{{d{b_B}}}{{dz}} = i({{{\tilde{K}}_{(a)(a)}} + {\beta_a}{C_{(a)(a)}}} )a(z) + }\\ { + i({\tilde{K}_{(bF)(a)}^{} + {\beta_a}C_{(a)(bF)}^{}} ){b_F}(z) + i({\tilde{K}_{(bB)(a)}^{} + {\beta_a}C_{(a)(bB)}^{}} ){b_B}(z),} \end{array}$$

with

(11)$$C_{(a)(j)}^{} = \frac{1}{4}\left( {\int { - E_x^{(j) + } \times H_y^{(a) - } + E_x^{(a) - } \times H_y^{(j) + }} } \right)dx,$$

(12)$$\tilde{K}_{(j)(a)}^{} ={-} \frac{\omega }{4}\int {\left( { - E_x^{(a) - }({\varepsilon_{xx}^{(a)}(x) - {\varepsilon_{xx}}(x)} )E_x^{(j) + } + E_z^{(a) - }\frac{{({\varepsilon_{zz}^{(a)}(x) - {\varepsilon_{zz}}(x)} )}}{{{\varepsilon_{zz}}(x)}}\varepsilon_{zz}^{(b)}(x)E_z^{(j) + }} \right)} .$$

where j = a, bF, bB.

2.2.2 Step 2

In order to obtain the second coupled-mode equation, as the system <2> with the field solutions (E⁽²⁾, H⁽²⁾) we choose the standalone THMM-based waveguide with permittivity spatial distribution given by Eq. (8), i.e., ${\overline{\bar{\varepsilon }}^{(2)}} = {\overline{\bar{\varepsilon }}^{(b)}}(x)$, and the field solution to be guided TM_0(bF) mode propagating in the -z direction with field components $\left( {\left( {\begin{array}{ccc} {E_x^{(bF) - },}&{0,}&{E_z^{(bF) - }} \end{array}} \right),\left( {\begin{array}{ccc} {0,}&{H_y^{(bF) - },}&0 \end{array}} \right)} \right)$. Thus, we have:

(13a)$${{\boldsymbol E}^{(2)}} = ({{\boldsymbol E}_t^{(bF) - } + {\boldsymbol E}_z^{(bF) - }} ){e^{ - i{\beta _{(bF)}}z}} = ({E_x^{(bF) - } \cdot \hat{x} + E_z^{(bF) - } \cdot \hat{z}} ){e^{ - i{\beta _{(bF)}}z}},$$

(13b)$${{\boldsymbol H}^{(2)}} = ({{\boldsymbol H}_t^{(bF) - } + {\boldsymbol H}_z^{(bF) - }} ){e^{ - i{\beta _{(bF)}}z}} = ({H_y^{(bF) - } \cdot \hat{y}} ){e^{ - i{\beta _{(bF)}}z}},$$

where β_(bF) is the propagation constant of the TM_0(bF) mode. It is worth noting that, the system <1> still remains the same as in the previous step, i.e., ${\overline{\bar{\varepsilon }}^{(1)}} = \overline{\bar{\varepsilon }}(x)$ [see Eq. (4)] and the set of fields $({{\boldsymbol E}^{(1)}},{{\boldsymbol H}^{(1)}})$ are given by Eqs. (5a)-(6b).

Again, by substituting $({{\boldsymbol E}^{(1)}},{{\boldsymbol H}^{(1)}})$, $({{\boldsymbol E}^{(2)}},{{\boldsymbol H}^{(2)}})$, ${\overline{\bar{\varepsilon }}^{(1)}}$, and ${\overline{\bar{\varepsilon }}^{(2)}}$ defined above into Eq. (3) we can obtain the following the second coupled-mode equation:

(14)$$\begin{array}{c} {{C_{(bF)(bF)}}\frac{{d{b_F}}}{{dz}} + {C_{(bF)(a)}}\frac{{da}}{{dz}} + {C_{(bF)(bB)}}\frac{{d{b_B}}}{{dz}} = i({{{\tilde{K}}_{(bF)(bF)}} + \beta_{(bF)}^{}{C_{(bF)(bF)}}} ){b_F}(z) + }\\ { + i({{{\tilde{K}}_{(a)(bF)}} + \beta_{(bF)}^{}{C_{(bF)(a)}}} )a(z) + i({{{\tilde{K}}_{(bB)(bF)}} + \beta_{(bF)}^{}{C_{(bF)(bB)}}} ){b_B}(z),} \end{array}$$

where

(15)$$C_{(bF)(j)}^{} = \frac{1}{4}\int {({ - E_x^{(j) + } \times H_y^{(bF) - } + E_x^{(bF) - } \times H_y^{(j) + }} )} dx,$$

(16)$${\tilde{K}_{(j)(bF)}} ={-} \frac{\omega }{4}\int {\left( { - E_x^{(bF) + }({\varepsilon_{xx}^{(b)}(x) - {\varepsilon_{xx}}(x)} )E_x^{(j) + } + E_z^{(bF) - }\frac{{({\varepsilon_{zz}^{(b)}(x) - {\varepsilon_{zz}}(x)} )}}{{{\varepsilon_{zz}}(x)}}\varepsilon_{zz}^{(a)}(x)E_z^{(j) + }} \right)dx} ,$$

where j = a, bF, bB.

2.2.3 Step 3

Finally, in order to obtain the third coupled-mode equation, we choose again the standalone THMM-based waveguide for the system <2> with the spatial distribution of permittivity ${\overline{\bar{\varepsilon }}^{(2)}} = {\overline{\bar{\varepsilon }}^{(b)}}(x)$, [see Eq. (8)], and the field solution $({{\boldsymbol E}^{(2)}},{{\boldsymbol H}^{(2)}})$ to be guided TM_0(bB) mode, this time propagating in the -z direction, with field components $\left( {\left( {\begin{array}{ccc} {E_x^{(bB) - },}&{0,}&{E_z^{(bB) - }} \end{array}} \right),\left( {\begin{array}{ccc} {0,}&{H_y^{(bB) - },}&0 \end{array}} \right)} \right)$. Thus, now (E⁽²⁾, H⁽²⁾) takes following form

(17a)$${{\boldsymbol E}^{(2)}} = ({{\boldsymbol E}_t^{(bF) - } + {\boldsymbol E}_z^{(bF) - }} ){e^{ - i{\beta _{(bF)}}z}} = ({E_x^{(bF) - } \cdot \hat{x} + E_z^{(bF) - } \cdot \hat{z}} ){e^{ - i{\beta _{(bF)}}z}},$$

(17b)$${{\boldsymbol H}^{(2)}} = ({{\boldsymbol H}_t^{(bB) - } + {\boldsymbol H}_z^{(bB) - }} ){e^{ - i{\beta _{(bB)}}z}} = ({H_y^{(bB) - } \cdot \hat{y}} ){e^{ - i{\beta _{(bB)}}z}},$$

where β_(bB) is the propagation constant of TM_0(bB) mode. Again, the system <1> remains the same as in previous steps.

By introducing (E⁽¹⁾, H⁽¹⁾) as in Eqs. (5) and (6), (E⁽²⁾, H⁽²⁾) as in Eq. (17), ${\overline{\bar{\varepsilon }}^{(1)}} = \overline{\bar{\varepsilon }}(x)$ and ${\overline{\bar{\varepsilon }}^{(2)}} = {\overline{\bar{\varepsilon }}^{(b)}}(x)$(described by Eq. (4) and Eq. (8), respectively) into Eq. (3), we obtain the third coupled-mode equation:

(18)$$\begin{array}{c} {{C_{(bB)(bB)}}\frac{{d{b_B}}}{{dz}} + {C_{(bB)(a)}}\frac{{da}}{{dz}} + {C_{(bB)(bF)}}\frac{{d{b_F}}}{{dz}} = i({{{\tilde{K}}_{(bB)(bB)}} + \beta_{(bB)}^{}{C_{(bB)(bB)}}} ){b_B}(z) + }\\ { + i({{{\tilde{K}}_{(a)(bB)}} + \beta_{(bB)}^{}{C_{(bB)(a)}}} )a(z) + i({{{\tilde{K}}_{(bF)(bB)}} + \beta_{(bB)}^{}{C_{(bB)(bF)}}} ){b_F}(z),} \end{array}$$

where

(19)$${C_{(bB)(bB)}} = \frac{1}{4}\int {({ - E_x^{(bB) + } \cdot H_y^{(bB) - } + E_x^{(bB) - } \cdot H_y^{(bB) + }} )} dx,$$

(20)$${\tilde{K}_{(bB)(bB)}} ={-} \frac{\omega }{4}\int {\left( { - E_x^{(bB) + }({\varepsilon_{xx}^{(b)}(x) - {\varepsilon_{xx}}(x)} )E_x^{(bB) + } + E_z^{(bB) - }\frac{{({\varepsilon_{zz}^{(b)}(x) - {\varepsilon_{zz}}(x)} )}}{{{\varepsilon_{zz}}(x)}}\varepsilon_{zz}^{(b)}(x)E_z^{(bB) + }} \right)dx} ,$$

where j = a, bF, bB.

The Eqs. (10), (14), and (18) are the coupled-mode equations describing intermodal coupling in the considered waveguide system, see Fig. 1(c), and can be written in the following form:

(21a)$$\frac{{da(z)}}{{dz}} = i{\gamma _a}a(z) + i{K_{(a)(bF)}}{b_F}(z) + i{K_{(a)(bB)}}{b_B}(z),$$

(21b)$$\frac{{d{b_F}(z)}}{{dz}} = i{\gamma _{bF}}{b_F}(z) + i{K_{(bF)(a)}}a(z) + i{K_{(bF)(bB)}}{b_B}(z),$$

(21c)$$\frac{{d{b_B}(z)}}{{dz}} = i{\gamma _{bB}}{b_B}(z) + i{K_{(bB)(a)}}a(z) + i{K_{(bB)(bF)}}{b_F}(z),$$

where γ_a_,bF,bB, are self-coupling coefficients of the TM _{0(a),(bF),(bB)} modes, while K_{(a)(bF),(bF)(a)}, K_{(a)(bB),(bB)(a)}, and K_{(bF)(bB),(bB)(bF)} are the cross-coupling coefficients that describe the strength of coupling between the corresponding modes. In particular, K_(a)(bF) is responsible for coupling of TM_0(bF) to TM_0(a), while the K_(bF)(a) corresponds to coupling of the TM_0(a) to TM_0(bF). Similarly, K_{(a)(bB),(bB)(a)} and K_{(bF)(bB),(bB)(bF)} are responsible for respective interactions between modes TM_0(a) and TM_0(bB), as well as TM_0(bF) and TM_0(bB), respectively. Explicit forms of coefficients γ_a_,bF,bB, K_{(a)(bF),(bF)(a)}, K_{(a)(bB),(bB)(a)} and K_{(bF)(bB),(bB)(bF)} are included in Appendix A1. It is worth noting, that coupling between two modes may occur directly or indirectly, i.e., via the third mode. For example, let us consider the coupling between TM_0(bF) and TM_0(bB). In this case, the direct coupling occurs according to the following scheme: $TM_{0(bB)} \mathop\rightleftarrows\limits_{{\rm K}_{(bB)(bF)}}^{{\rm K}_{(bF)(bB)}} TM_{0(bF)}$, while the indirect coupling may be described as follows: $T{M_{0(bB)}} \mathop\rightleftarrows\limits_{{\rm K}_{(bB)(a)}}^{{\rm K}_{(a)(bB)}} T{M_{0(a)}} \mathop\rightleftarrows\limits_{{\rm K}_{(a)(bF)}}^{{\rm K}_{(a)(bF)}} T{M_{0(bF)}}$.

For clarity and convenience, one may rewrite Eq. (21) in matrix form:

(22)$$\frac{d}{{dz}}\left[ {\begin{array}{c} {a(z)}\\ \begin{array}{l} {b_F}(z)\\ {b_B}(z) \end{array} \end{array}} \right] = \left[ {\begin{array}{ccc} {i{\gamma_a}}&{i{K_{(a)(bF)}}}&{i{K_{(a)(bB)}}}\\ {i{K_{(bF)(a)}}}&{i{\gamma_{(bF)}}}&{i{K_{(bF)(bB)}}}\\ {i{K_{(bB)(a)}}}&{i{K_{(bB)(bF)}}}&{i{\gamma_{(bB)}}} \end{array}} \right]\left[ {\begin{array}{c} {a(z)}\\ \begin{array}{l} {b_F}(z)\\ {b_B}(z) \end{array} \end{array}} \right],$$

Based on solutions of Eq. (22), i.e., a(z), b_F(z), and b_B(z), with appropriate initial conditions, it is possible to calculate variation of the optical power of each mode propagating in + z direction, caused by intermodal coupling in the considered waveguide system, in the following way:

(23a)$$\begin{array}{c} {{P_a}(z) = \frac{1}{2}Re \left\{ {|{a{{(z)}^2}} |\int {E_x^{(a)} \cdot H_y^{ {\ast} (a)}dx + } } \right.}\\ {\left. { + {a^ \ast }(z){b_F}(z)\int {E_x^{(bF)} \cdot H_y^{ {\ast} (a)}dx} - {a^ \ast }(z){b_B}(z)\int {E_x^{(bB)} \cdot H_y^{ {\ast} (a)}dx} } \right\}} \end{array},$$

(23b)$$\begin{array}{c} {{P_{(bF)}}(z) = \frac{1}{2}Re \left\{ {|{{b_F}{{(z)}^2}} |\int {E_x^{(bF)} \cdot H_y^{ {\ast} (bF)}dx + } } \right.}\\ { + b_F^\ast (z)a(z)\left. {\int {E_x^{(a)} \cdot H_y^{ {\ast} (bF)}dx} - b_F^\ast (z){b_B}(z)\int {E_x^{(bB)} \cdot H_y^{ {\ast} (bF)}dx} } \right\}} \end{array},$$

(23c)$$\begin{array}{c} {{P_{(bB)}}(z) = \frac{1}{2}Re \left\{ {|{{b_B}{{(z)}^2}} |\int {E_x^{(bB)} \cdot H_y^{ {\ast} (bB)}dx} } \right.}\\ {\left. { - b_B^\ast (z)a(z)\int {E_x^{(a)} \cdot H_y^{ {\ast} (bB)}dx} - b_B^\ast (z){b_F}(z)\int {E_x^{(bF)} \cdot H_y^{ {\ast} (bB)}dx} } \right\}} \end{array}.$$

It is worth to underline that throughout our analysis, all the field distributions of TM_0(a), TM_0(bF), and TM_0(bB) modes are normalized to the unit power.

3. Results and discussion

In this section we present numerical solutions to the coupled mode equations, see (Eq. (2)), for a waveguide system composed of an isotropic dielectric waveguide and an anisotropic THMM waveguide, see Fig. 1(c), with wavelength fixed at λ₀=1.55 µm. The core layer of the isotropic dielectric waveguide is composed of boron doped silicon dioxide (BSG) of ɛ_a = 2.1 [52]. The core layer of the anisotropic THMM waveguide is formed by periodically arranged bilayers composed of six graphene monolayers, with dielectric permittivity described by well-known Kubo formula [53], and 2 nm layer of BSG (ɛ_d = 2.1 [52]). In our approach, the THMM waveguide core is described as a uniaxial anisotropic medium of permittivity tensor ${\overline{\bar{\varepsilon }}_b} = diag\left( {\left[ {\begin{array}{ccc} {{\varepsilon_\parallel },}&{{\varepsilon_\parallel },}&{{\varepsilon_ \bot }} \end{array}} \right]} \right) = diag\left( {\left[ {\begin{array}{ccc} {{\varepsilon_{xx}},}&{{\varepsilon_{yy}},}&{{\varepsilon_{zz}}} \end{array}} \right]} \right)$ with components obtained with the help of the effective medium theory (EMT) [40]. Moreover, to ensure the existence of Type II hyperbolic dispersion for the wavelength considered, we designed the unit cell of the HMM structure as described above and assumed that chemical potential of graphene is higher than 0.6 eV [42,46]. Finally, the core layer is cladded by silicon dioxide with lower level of boron doping, exhibiting relative electric permittivity ɛ_c = 2.095 [52].

It is worth to underline that all observed propagation properties, and consequently the intermodal coupling, are not distinctive features of the constituent materials. Similar effects may be obtained in any system with a waveguide revealing controllable Type II hyperbolic dispersion. However, it is still purposeful to briefly discuss the technological feasibility of the proposed system. The materials constituting the unit cell of the THMM medium, i.e., graphene and BSG, of chosen thicknesses, can be manufactured via well-established technology, such as magnetron sputtering [52], epitaxial growth or chemical vapor deposition [54]. However, to manufacture the complete multilayer THMM structure, it is required to combine multiple technological processes, including transfer of graphene layers [55]. Until now, it has been demonstrated that similar graphene-based multilayer structures have been successfully assembled [55]. It is worth noting that, the fabrication of the proposed waveguide system may require use of additional technological processes, such as etching and/or lithography. Thus, the considered waveguide system is feasible by means of compatible technological processes.

To investigate intermodal coupling in the two-waveguide system of parameters defined above, we must first determine propagation properties of individual waveguides constituting the system. As previously indicated, our analysis is focused on intermodal coupling between TM modes. Thus, the dispersion characteristics for TM modes propagating in the isotropic dielectric and the THMM waveguides have been illustrated in Figs. 2(a) and 2(b), respectively. For the wavelength λ₀=1.55µm and the material parameters assumed, the dispersion curves have been calculated using well-established characteristic equations of a isotropic dielectric waveguide [3–5] and an anisotropic HMM waveguide described with effective permittivity tensor [45,46]. Moreover, since the employed constituent materials are considered as low-loss in the considered spectral range [42,46,52], without loss of the generality, we can limit our analysis to real part of the propagation constant. As we can see in Fig. 2(a), the dielectric waveguide may support only fundamental TM_0(a) mode if the core layer width is lower than approximately 10 µm. In the case of THMM waveguide, for a given waveguide width, two modes of the same order and the same propagation direction having different propagation constants and opposite signs of power flow are supported, see Fig. 2(b). The lower branches (solid lines) of the dispersion curves correspond to forward modes TM_n(bF) having power flow coinciding with the direction of propagation, while the upper branches (dashed lines) are related to the backward modes TM_n(bB) with the power flow opposite to the propagation direction. By choosing a THMM waveguide width as b=0.088 µm, indicated in Fig. 2(b) as a vertical dashed grey line, we ensure guidance conditions for a single fundamental forward mode TM_0(bF) but still an infinite number of backward TM_n(bB) modes. Moreover, similar guidance conditions are maintained for chemical potential of graphene higher than 0.6 eV. It is worth noting, that in general the appropriate level of graphene’s chemical potential may be provided by means of chemical doping [56], while the active tuning can be performed via change of temperature as well as external magnetic or electric field [53,56]. In particular, the electrical gating in the practical implementation of waveguide system may be realized in the form of capacitor-like scheme, where a THMM/HMM waveguide is sandwiched between conductive substrate and top electrode serving as ohmic contacts. Furthermore, by comparing the dispersion characteristics presented in Figs. 2(a) and 2(b) one can notice that for a given chemical potential and THMM waveguide width, the propagation constants of the higher-order TM_n(bB) modes differ substantially from the propagation constants of the fundamental modes TM_0(a), TM_0(bF), and TM_0(bB). Thus, we can expect that the significant mode coupling will occur only between TM_0(a), TM_0(bF), and TM_0(bB) modes and the coupling with the higher-order TM_n(bB) modes can be neglected

Fig. 2. Propagation constants for TM guided modes of (a) dielectric and (b) THMM-based waveguide plotted vs. waveguide width.

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Now, we numerically solve the coupled-mode equations, Eq. (22) with initial conditions a(z = 0) = 1 and b_F(z = 0) = b_B(z = 0) = 0, i.e., the unit optical power is launched into the dielectric waveguide, to further calculate - with the help of Eq. (23) - the power in each of the considered modes as a function of the perturbation length, i.e., as function of z, for systems of various dielectric waveguide widths a as well as separation distances d between the waveguide core layers. The considered range of dielectric waveguide widths, i.e., up to 10µm, ensures single mode operation (propagation of the fundamental mode TM_0(a) only), see Fig. 2(a). Within the following analysis we also assume that chemical potential of graphene equals to µ_C = 0.6 eV and we set the THMM waveguide width as b=0.088µm, see Fig. 2(b).

Figures 3(a)–3(f) illustrate behavior of optical power P_a, P_bF, P_bB of the modes TM_0(a), TM_0(bF), and TM_0(bB), respectively, varying with position z (i.e., as a function of the perturbation length) for various dielectric waveguide widths a, Figs. 3(a)–3(c), and different separation distances d between the waveguide core layers, Figs. 3(d)–3(f). In general, the power transfer between the TM_0(a) of the dielectric waveguide and forward mode TM_0(bF) as well as backward mode TM_0(bB) of the THMM waveguide is possible and total power propagating in the waveguide system is constant and equal to the optical power initially launched into the dielectric waveguide.

Fig. 3. The modal optical power fluctuations vs. position z for the (a,b) TM_0(a), (c,d) TM_0(bF) and (e,f) TM_0(bB) for different dielectric waveguide widths a (a,b,c) and various separation distances d (d,e,f).

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By comparing Figs. 3(a)–3(c), it can be noticed that intermodal coupling depends on the width of the dielectric waveguide for the fixed separation distance d between the waveguide core layers and the assumed THMM waveguide width. There exists a dielectric waveguide width (in this case a = 4 µm), for which the intermodal power transfer between modes of the dielectric and the THMM waveguide is the most efficient.

Now, with the dielectric waveguide fixed at width a = 4 µm, we investigate influence of the separation distance between the core layers d on the coupling efficiency, see Figs. 3(d)–3(f). As expected, decreasing the separation distance leads to stronger intermodal power transfer between the mode of the dielectric waveguide, i.e., TM_0(a), and the modes of the THMM waveguide, i.e., TM_0(bF) and TM_0(bB). This is consistent with intermodal coupling observed in typical coupled-waveguide systems [57], since decrease of the separation between waveguides leads to higher field overlap between the modes propagating in the waveguides.

Since the chemical potential µ_c of graphene can be externally controlled, for example by the change of temperature, external magnetic field or external voltage [53,56]. By varying one of these external stimuli, it is possible to control propagation properties of the THMM waveguide, and as consequence, mode coupling within the system. The respective characteristics revealing the dependence of the intermodal power transfer on the chemical potential µ_c are presented in Figs. 4(a)–4(d).

Fig. 4. The optical power changes of TM_0(a), TM_0(bF) and TM_0(bB) modes plotted vs. position z for different chemical potentials of graphene µ_C (a) 0.6 eV, (b) 0.65 eV, (c) 0.7 eV, and (d) 1 eV.

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For chosen parameters and µ_C = 0.6 eV, the power transfer between dielectric, i.e., TM_0(a) mode, and THMM waveguide, i.e., backward TM_0(bB) and forward TM_0(bF) modes, is observed, see Fig. 4(a). It is worth noting that, in this case, the mismatch between propagation constants of the considered modes is the lowest, compare Figs. 2(a) and 2(b). As we can see, increasing chemical potential of graphene leads to weakening of the power transfer to the THMM backward mode TM_0(bB), which is caused by an increasing mismatch between propagation constants. Simultaneously, in the case of TM_0(bF) mode, higher values of chemical potential lead to the weaker power transfer, which, on the other hand, is caused by suppression of indirect coupling through the backward mode TM_0(bB), see Figs. 4(a) and 4(b). However, further increase of chemical potential provides stronger coupling from the dielectric waveguide mode TM_0(a) to the forward THHM waveguide mode TM_0(bF), see Figs. 4(c) and 4(d), since the propagation-constant mismatch becomes smaller, compare Figs. 2(a) and 2(b). Moreover, the change of chemical potential enables shift of the perturbation length for which the power transfer reaches its maximum, see Figs. 4(a)–4(d).

Thus, by changing chemical potential, it is possible to control strength of power exchange between chosen modes. As an example, we present two representative cases corresponding to two possible scenarios of controlling power transfer in the considered waveguide system, see Figs. 5(a)–5(b). As we can see, by choosing an appropriate perturbation length z (coupling length), it is possible to obtain new means for controlling propagation in the waveguide system, including enhancing or suppressing power transfer to chosen mode. In the first scenario, we start from equal power transfer to both forward and backward THMM waveguide mode, i.e., TM_0(bF) and TM_0(bB), which correspond to equal power flow in both + z and -z direction, and, by increasing chemical potential, it is possible to suppress transfer to the TM_0(bB), and consequently negligible power flow in the -z direction, see Fig. 5(a). The second scenario provides a possibility to control coupling between the dielectric waveguide mode TM_0(a) and chosen modes of the THMM waveguide TM_0(bB) and TM_0(bF). In particular, with a proper choice of chemical potential, it is possible to obtain coupling solely to TM_0(bB), and thus the power flow in the -z direction only, or coupling to TM_0(bF) only, and thus power flow in the + z direction only. Moreover, it is also possible to achieve simultaneous coupling to both TM_0(bB) and TM_0(bF) modes, see Fig. 5(b). Therefore, it is possible to dynamically control intermodal power transfer in the THMM-based waveguide system, including changing the direction of power flow at will.

Fig. 5. The intermodal power fluctuations for two chosen perturbation lengths z: (a) 2.35 µm and (b) 48.8 µm plotted as a function of µ_C.

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4. Conclusions

For the first time, we have demonstrated an analysis of controlled intermodal power transfer in a hybrid THMM-dielectric waveguide system. In particular, we have presented an explicit derivation of coupled mode equations based on the generalized reciprocity relation (i.e., the Lorentz theorem), which served to demonstrate the possibility of power transfer between the TM modes of the dielectric waveguide and forward as well as backward TM modes of the THMM waveguide. It is worth to underline that, according to our knowledge, this work is the first approach to investigate intermodal coupling in a waveguide system based on HMM/THMM medium. Within the course of our analysis we have determined effects of characteristic parameters of the system, such as separation between core layer and their widths, on the intermodal power transfer. Moreover, we have also demonstrated for the first time that the change of the graphene’s chemical potential (e.g., by external biasing) enables control of the coupling strength between the chosen modes, providing tunable power transfer with controllable power-flow direction. Finally, we have presented, as examples, two possible scenarios for controlling propagation and intermodal power transfer in the considered system. We believe that the theoretical results presented in this paper will serve as a basis for future practical implementation and further studies exploring possibilities arising from incorporation of HMM/THMM into optical integrated circuits.

Appendix A1

Longitudinal field components

Starting from Maxwell’s equations, the longitudinal field components can be expressed in terms of the transverse field components in the following manner:

(24a)$${\boldsymbol E}_z^{(1)} = \frac{i}{{\omega {\varepsilon _{zz}}}}\{{{\nabla_t} \times {\boldsymbol H}_t^{(1)} + i\omega {{\overline{\bar{\varepsilon }}}_{zt}} \cdot {\boldsymbol E}_t^{(1)}} \},$$

(24b)$${\boldsymbol H}_z^{(1)} = \frac{1}{{i\omega \mu }}{\nabla _t} \times {\boldsymbol E}_t^{(1)}.$$

Since the waveguide system considered (regarded as system <1>) is characterized by a diagonal dielectric tensor ${\overline{\bar{\varepsilon }}^{(1)}}$, ${\overline{\bar{\varepsilon }}_{zt}}$ equals zero and Eq. (24a) takes the form:

(25)$${\boldsymbol E}_z^{(1)} = \frac{i}{{\omega {\varepsilon _{zz}}}}{\nabla _t} \times {\boldsymbol H}_t^{(1)}. $$

Inserting into (25) the transverse component ${\boldsymbol H}_t^{(1)}$ as a superposition of individual guided modes, we obtain:

(26)$${\boldsymbol E}_z^{(1)} = \frac{i}{{\omega {\varepsilon _{zz}}}}\{{a(z){\nabla_t} \times {\boldsymbol H}_t^{(a) + } + {b_F}(z){\nabla_t} \times {\boldsymbol H}_t^{(bF) + } + {b_B}(z){\nabla_t} \times {\boldsymbol H}_t^{(bB) + }} \}. $$

Next, using Eq. (25) for the guided modes, we have:

(27a)$${\nabla _t} \times {\boldsymbol H}_t^{(a) + } ={-} i\omega \varepsilon _{_{zz}}^{(a)}{\boldsymbol E}_z^{(a) + },$$

(27b)$${\nabla _t} \times {\boldsymbol H}_t^{(bF) + } ={-} i\omega \varepsilon _{_{zz}}^{(b)}{\boldsymbol E}_z^{(bF) + },$$

(27c)$${\nabla _t} \times {\boldsymbol H}_t^{(bB) + } ={-} i\omega \varepsilon _{_{zz}}^{(b)}{\boldsymbol E}_z^{(bB) + }.$$

Inserting formulas (27) into (26) gives:

(28)$${\boldsymbol E}_z^{(1)} = \left( {\frac{{a(z)}}{{{\varepsilon_{zz}}}}\varepsilon_{zz}^{(a)}{\boldsymbol E}_z^{(a) + } + \frac{{{b_F}(z)}}{{{\varepsilon_{zz}}}}\varepsilon_{zz}^{(b)}{\boldsymbol E}_z^{(bF) + } + \frac{{{b_B}(z)}}{{{\varepsilon_{zz}}}}\varepsilon_{zz}^{(b)}{\boldsymbol E}_z^{(bB) + }} \right),$$

which corresponds to Eq. (6a) in Section 2.2:

(29)$${\boldsymbol E}_z^{(1)} = E_z^{(1)} \cdot \hat{z} = \left( {\frac{{a(z)}}{{{\varepsilon_{zz}}}}\varepsilon_{zz}^{(a)}E_z^{(a) + }(x) + \frac{{{b_F}(z)}}{{{\varepsilon_{zz}}}}\varepsilon_{zz}^{(b)}E_z^{(bF) + }(x) + \frac{{{b_B}(z)}}{{{\varepsilon_{zz}}}}\varepsilon_{zz}^{(b)}E_z^{(bB) + }(x)} \right) \cdot \hat{z}.$$

A similar procedure can be applied to the longitudinal component ${\boldsymbol H}_z^{(1)}$.

Appendix A2

Coupled-mode-equation coefficients

Exact analytical expressions for all the coefficients occurring in Eqs. (21) and (22) are the following:

(30)$${\gamma _a} = \frac{{({{\alpha_1}C + {\alpha_3}{{\bar{C}}_2}} )({{C^2} + C_{12}^2} )- ({{\alpha_2}C + {\alpha_3}{C_{12}}} )({{{\bar{C}}_1}C + {C_{12}}{{\bar{C}}_2}} )}}{{({{C^2} + \bar{C}_2^2} )({{C^2} + C_{12}^2} )- {{({{{\bar{C}}_1}C + {C_{12}}{{\bar{C}}_2}} )}^2}}},$$

(31)$${K_{(a)(bF)}} = \frac{{({{\gamma_1}C + {\gamma_3}{{\bar{C}}_2}} )({{C^2} + C_{12}^2} )- ({{\gamma_2}C + {\gamma_3}{C_{12}}} )({{{\bar{C}}_1}C + {C_{12}}{{\bar{C}}_2}} )}}{{({{C^2} + \bar{C}_2^2} )({{C^2} + C_{12}^2} )- {{({{{\bar{C}}_1}C + {C_{12}}{{\bar{C}}_2}} )}^2}}},$$

(32)$${K_{(a)(bB)}} = \frac{{({{\theta_1}C + {\theta_3}{{\bar{C}}_2}} )({{C^2} + C_{12}^2} )- ({{\theta_2}C + {\theta_3}{C_{12}}} )({{{\bar{C}}_1}C + {C_{12}}{{\bar{C}}_2}} )}}{{({{C^2} + \bar{C}_2^2} )({{C^2} + C_{12}^2} )- {{({{{\bar{C}}_1}C + {C_{12}}{{\bar{C}}_2}} )}^2}}},$$

(33)$${\gamma _{bF}} = \frac{{({{\gamma_3}{C_{12}} + {\gamma_2}C} )({C{C_{12}} + {{\bar{C}}_1}{{\bar{C}}_2}} )- ({{\gamma_1}{C_{12}} - {\gamma_2}{{\bar{C}}_2}} )({{{\bar{C}}_2}{C_{12}} + {{\bar{C}}_1}C} )}}{{({{C^2} + \bar{C}_{12}^2} )({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )- ({{{\bar{C}}_1}{C_{12}} - C{{\bar{C}}_2}} )({{{\bar{C}}_2}{C_{12}} + {{\bar{C}}_1}C} )}},$$

(34)$${K_{(bF)(a)}} = \frac{{({{\alpha_3}{C_{12}} + {\alpha_2}C} )({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )- ({{\alpha_1}{C_{12}} - {\alpha_2}{{\bar{C}}_2}} )({{{\bar{C}}_2}{C_{12}} + {{\bar{C}}_1}C} )}}{{({{C^2} + \bar{C}_{12}^2} )({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )- ({{{\bar{C}}_1}{C_{12}} - C{{\bar{C}}_2}} )({{{\bar{C}}_2}{C_{12}} + {{\bar{C}}_1}C} )}},$$

(35)$${K_{(bF)(bB)}} = \frac{{({{\theta_3}{C_{12}} + {\theta_2}C} )({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )- ({{\theta_1}{C_{12}} - {\theta_2}{{\bar{C}}_2}} )({{{\bar{C}}_2}{C_{12}} + {{\bar{C}}_1}C} )}}{{({{C^2} + \bar{C}_{12}^2} )({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )- ({{{\bar{C}}_1}{C_{12}} - C{{\bar{C}}_2}} )({{{\bar{C}}_2}{C_{12}} + {{\bar{C}}_1}C} )}},$$

(36)$${\gamma _{bB}} = \frac{{({{\theta_3}C - {\theta_1}{{\bar{C}}_2}} )({{C^2} - \bar{C}_1^2} )- ({{\theta_2}C - {\theta_1}{{\bar{C}}_1}} )({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )}}{{ - ({{C^2} + \bar{C}_2^2} )({{C^2} - \bar{C}_1^2} )- {{({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )}^2}}},$$

(37)$${K_{(bB)(a)}} = \frac{{({{\alpha_3}C - {\alpha_1}{{\bar{C}}_2}} )({{C^2} - \bar{C}_1^2} )- ({{\alpha_2}C - {\alpha_1}{{\bar{C}}_1}} )({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )}}{{ - ({{C^2} + \bar{C}_2^2} )({{C^2} - \bar{C}_1^2} )- {{({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )}^2}}},$$

(38)$${K_{(bB)(bF)}} = \frac{{({{\gamma_3}C - {\gamma_1}{{\bar{C}}_2}} )({{C^2} - \bar{C}_1^2} )- ({{\gamma_2}C - {\gamma_1}{{\bar{C}}_1}} )({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )}}{{ - ({{C^2} + \bar{C}_2^2} )({{C^2} - \bar{C}_1^2} )- {{({C{C_{12}} - {{\bar{C}}_1}{{\bar{C}}_2}} )}^2}}},$$

where:

(39a-c)$${\alpha _1} = \begin{array}{ccc} {{{\tilde{K}}_{aa}} + {\beta _a}C,}&{{\alpha _2} = {{\tilde{K}}_{(a)(bF)}} + {\beta _{bF}}{{\bar{C}}_1},}&{{\alpha _3} = {{\tilde{K}}_{(a)(bB)}} + {\beta _{(bB)}}{{\bar{C}}_2}} \end{array},$$

(40a-c)$$\begin{array}{ccc} {{\gamma _1} = {{\tilde{K}}_{(bF)(a)}} + {\beta _a}{{\bar{C}}_1},}&{{\gamma _2} = {{\tilde{K}}_{(bF)(bF)}} + {\beta _{(bF)}}C,}&{{\gamma _3} = {{\tilde{K}}_{(bF)(bB)}} + {\beta _{(bB)}}{C_{12}}} \end{array},$$

(41a-c)$$\begin{array}{ccc} {{\theta _1} = {{\tilde{K}}_{(bB)(a)}} + {\beta _a}{{\bar{C}}_2},}&{{\theta _2} = {{\tilde{K}}_{(bB)(bF)}} + {\beta _{(bF)}}{C_{12}},}&{{\theta _3} = {{\tilde{K}}_{(bB)(bB)}} + {\beta _{(bB)}}C,} \end{array}$$

(42a-b)$${C_{(a)(a)}} = {C_{(bF)(bF)}} = C$$

(43a-c)$$\begin{array}{ccc} {{{\bar{C}}_1} = {C_{(a)(bF)}} = {C_{(bF)(a)}},}&{{{\bar{C}}_2} = {C_{(a)(bB)}} = {C_{(bB)(a)}},}&{{C_{12}} = {C_{(bF)(bB)}} = {C_{(bB)(bF)}}} \end{array}.$$

Funding

Politechnika Warszawska (Excellence Initiative: research University (ID-UB)).

Acknowledgments

Studies were funded by Materials Technologies project granted by Warsaw University of Technology under the program Excellence Initiative: Research University (ID-UB).

Disclosures

The authors declare no conflicts of interest.

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Controllable intermodal coupling in waveguide systems based on tunable hyperbolic metamaterials

Abstract

1. Introduction

2. Theoretical model

2.1 Considered system

2.2 Coupled-mode equations

2.2.1 Step 1

2.2.2 Step 2

2.2.3 Step 3

3. Results and discussion

4. Conclusions

Appendix A1

Longitudinal field components

Appendix A2

Coupled-mode-equation coefficients

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (5)

Equations (56)

Optics Express

Anna Tyszka-Zawadzka	https://orcid.org/0000-0002-6411-0570
Bartosz Janaszek	https://orcid.org/0000-0003-3719-2598
Marcin Kieliszczyk	https://orcid.org/0000-0001-9712-4162
Paweł Szczepański	https://orcid.org/0000-0002-2509-8724