1. Introduction

It is well-accepted that there are beautiful symmetric structures embedded in Maxwell’s equations, i.e., the dual symmetry between electric and magnetic field, time reversal symmetry, and many others as explained in [1]. Those symmetries on one hand could be used to simplify our understanding of mode hybridization associated with complicated optical structures [2], on the other hand impose certain constraints to electromagnetic response [3]. One also notes that certain optical structures based on combined symmetries of parity and time reversal $\mathcal{P}\mathcal{T}$ posses interesting features, i.e., real eigenvalues though the Hamiltonian being non-Hermitian, and exceptional points (EPs) where the transition of $\mathcal{P}\mathcal{T}$ symmetry breaking occurs. It is necessary to study and understand the general scenarios where those symmetries can be broken, leading to astonishing behaviors of light such as non-reciprocal or one-way propagation. In waveguides, there is an additional symmetry, i.e., translation symmetry along the propagation direction. Such translation symmetry ensures the modal wave number a constant value, i.e., propagation constant β, which is a typical terminology in waveguides. In analogy of a waveguide mode E(r) = e(x, y)e^−iβz to a wave-function Ψ(r, t) = Ψ(r)e^−iEt associated with the stationary Schrödinger equation HΨ(r) = EΨ(r), z plays the the role of time t, and β plays the role of energy E [4, 5].

In isotropic waveguide, the backward propagating modes (−β) can be considered as perfect images of the forward propagating modes (β). Importantly, such symmetric relation has been used extensively in coupled mode theory (CMT) [6], where the forward and backward propagating modes together establish the dual mode sets that are essential in CMT. Thus, it is interesting to ask how the forward and backward propagating modes are related in the context of CMT, for general reciprocal waveguides that may contain gain/losses, anisotropy, or bianisotropy? One major goal of this paper is to provide and classify such symmetric relation that connects forward and backward propagating modes, if such symmetry exists in general reciprocal waveguides.

Secondly, one also notes that if the waveguide cross-section contains rotational symmetry, the polarization modes associated with the same field configuration may or may not degenerate. Though those results are well documented in the literatures, there is no systematic approach to classify the symmetry properties of waveguide mode using the equivalent Hamiltonian, considering the analogy of the wave equation of the waveguides with the stationary Schrödinger equation. Along this line, it is important to point out the field of the waveguide mode is vector field, in contrast to the scalar wave function in Schrödinger equation.

In this work, we derive the exact Hamiltonian of the waveguide from Maxwell’s equations. In our formulation, we take into account the vectorial nature of electromagnetic field in the equivalent waveguide Hamiltonian, which resembles Dirac equation accounting for electrons with positive/negative energies and up/down spin states. By construction, we search for the symmetry operation associated with the Hamiltonian to classify the symmetry properties of waveguide modes in presence of gain/losses, anisotropy/bianisotropy, or continuous/discrete rotational symmetry in the geometric cross-section of the waveguides.

The paper is organized as follows: In Section 2, we outline the construction of Hamiltonian for the waveguide as well as the description of the symmetry operations. In Section 3, we apply the symmetry operations to the Hamiltonian of different waveguides, and to classify the symmetric properties of the waveguide modes. Finally, Section 4 concludes the paper.

2. Theory

2.1. Waveguide Hamiltonian

In general bianisotropic waveguide, the constitutive relation can be given as follows [7]:

Corresponding to the Hamiltonian in 4 × 4 matrix form, there will be 4 eigenmodes ${\mathrm{\Psi }}_1^{+}$, ${\mathrm{\Psi }}_2^{+}$, ${\mathrm{\Psi }}_1^{-}$ and ${\mathrm{\Psi }}_2^{+}$ in the truncated mode set, with eigenvalue being ${\beta }_1^{+}$, ${\beta }_2^{+}$, ${\beta }_1^{-}$ and ${\beta }_2^{+}$ respectively. The superscript +(−) indicates forward (backward) propagating modes, and we note a pair of orthogonal polarization modes in same direction with subscript 1 or 2. Once the waveguide Hamiltonian is known, the degeneracy of the modes within the truncated mode set can be classified by searching proper symmetry operations.

In the paper, we concern waveguiding mainly by the refractive index contrast. Thus, the waveguide can be sliced into regions with piece-wise constant material properties. To perform modal analysis of waveguide, one finds the eigenfield of each region, and then apply the boundary condition to connect the field from different regions such that the eigenfield of the waveguide can be obtained. This procedure shows that the final eigenfield of the waveguide can be seen as certain combination of the eigenfield of each individual region, though the boundary condition determines how the eigenfield from different region is combined. In any case, the final eigenfield of the waveguide mode obeys the same symmetry as the eigenfield of each individual region, provided the same modal wave number β is selected. Thus, the study on the symmetry properties of the waveguide mode can be reduced to analysis the symmetry properties of the eigenfied of each individual region, with no need to concern the boundary conditions. In our settings, the background of the waveguide core is air, the symmetry relation of the waveguide mode is essentially determined by the waveguide core, which is our focus in the following sections.

2.2. Chiral symmetry

We study the degeneracy between opposite propagating modes (${\mathrm{\Psi }}_1^{+}$ and${\mathrm{\Psi }}_1^{-}$ or ${\mathrm{\Psi }}_2^{+}$ and ${\mathrm{\Psi }}_2^{-}$). Here, an unitary matrix

2.3. Time reversal symmetry

Next, we introduce the time reversal operator $\mathcal{T}:\widehat{p}\to -\widehat{p}$, i ⇒ −i, where $\widehat{p}$ is the momentum operator [8, 9]. In general, this operator can be represented as $\mathcal{T}=UK$, where U is a unitary matrix and K is complex conjugation [10]. The operator σ used in chiral symmetry operation is an unitary matrix, and will be used here to replace U, leading to the time reversal operator as follows,

2.4. Parity symmetry

We proceed to discuss the symmetry operation that changes the coordinates, for example, parity operator $\mathcal{P}$, r → −r, $\widehat{p}\to -\widehat{p}$, where r is the position operator and only contains transverse coordinate (x, y) at here [8, 9]. The optical properties of waveguide are essentially determined by the spatial dependent permittivity and permeability, i.e., ${\overline{\mathit{\epsilon }}}_r(\mathit{r})$ and${\overline{\mathit{\mu }}}_r(\mathit{r})$. Considering $\overline{\mathit{\chi }}=0$, Eq. (2) can be reformulated as,

2.5. $\mathcal{P}\mathcal{T}$ symmetry

In the time reversal/parity symmetry operation, we have proved there is a definite relation between the forward and backward propagating modes that is guaranteed by $\mathcal{P}/\mathcal{T}$ symmetry. In this subsection, we continue to discuss the the symmetric properties induced by combining the two symmetry operations together, i.e., $\mathcal{P}\mathcal{T}$ symmetry, which has been examined extensively in the last few years [5, 6, 8, 9, 11]. As the operator $\mathcal{P}$ and $\mathcal{T}$ both act on Hamiltonian H, one obtains $\mathcal{P}\mathcal{T}H\left(\widehat{p},\mathit{r},t\right){\left(\mathcal{P}\mathcal{T}\right)}^{-1}=H^{\ast }\left(\widehat{p},-\mathit{r},-t\right)$. If the optical systems are $\mathcal{P}\mathcal{T}$ symmetric (here we only concerns isotropic medium), i.e., ${\epsilon }_r\left(\mathit{r}\right)={\epsilon }_r^{*}\left(-\mathit{r}\right)$, ${\mu }_r\left(\mathit{r}\right)={\mu }_r^{*}\left(-\mathit{r}\right)$, one find the waveguide Hamiltonian H commutes with the $\mathcal{P}\mathcal{T}$ operator, i.e., $\mathcal{P}\mathcal{T}H{\left(\mathcal{P}\mathcal{T}\right)}^{-1}=H$, leading to,

Table 1. Symmetry properties of waveguide modes in the truncated mode set

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2.6. Rotation and mirror symmetry

We continue to study the degeneracy between the polarization states Ψ₁, Ψ₂ due to the rotation or mirror symmetry of the cross-section of waveguides. To this end, we first discuss the difference between the rotation of vector field and that of scalar field. The transverse electromagnetic field components, which are the eigenfunction Ψ of waveguide Hamiltonian, is essentially a vector field. Thus, it is necessary to give formal expressions to describe how the vector and scalar field are rotated. According to [12–14], as a rotating operator O_R acts on a scalar field (for example, x component of electric field e_x) and a vector field (for example, transverse electric field ${\mathit{e}}_t={\left(e_x,e_y\right)}^T$), one shall have,

Considering the structure symmetry of the cross-section can be encoded into the optical properties of the material, we use Eq. (8) that explicitly encloses the coordinate-dependent material tensors, i.e., ${\overline{\mathit{\epsilon }}}_r\left(\mathit{r}\right)$ and ${\overline{\mathit{\mu }}}_r\left(\mathit{r}\right)$. For simplicity, we only consider isotropic waveguides such as ordinary optical fiber. To this end, the Hamiltonian H can be reduced as,

In analogy with the rotating operator O_R, the mirror reflection operator O_S [14] reflects a vector field by using reflection operation that acts on the coordinate and polarization,

2.7. Combinations of symmetry operations

To get a comprehensive impression of symmetry operations discussed in this paper, we list the six different symmetry operations discussed in previous sections in Table. 1. The first three symmetry operations are used to establish the symmetric relation between the forward and backward propagating modes, and the last three symmetry operations establish the relationship in the same propagating direction. In Section 2.2 and 2.3, we only the reshuffle the components of the vector field without touching on the coordinates. Thus, the operator σ in chiral symmetry and the operator T in time reversal symmetry essentially belongs to first step of transforming a vector field. In Sections 2.4, 2.5 and 2.6, those symmetry operations can be considered as combined operations of two steps that complete the full operation of transforming a vector field.

Evidently, the aforementioned parity symmetry is the combination of chiral symmetry and rotation symmetry with θ = π or the combination of chiral symmetry and mirror symmetries with θ = 0 and π. The $\mathcal{P}\mathcal{T}$ symmetry is the combination of time reversal symmetry and parity symmetry. Since some of the tabulated symmetry operations in Table. 1, i.e., chiral/rotation/mirror/time reversal symmetry, are elementary, we can combine two of them to get new symmetry operations. For example, the mirror symmetry can be combined with the chiral symmetry to get the degenerate between opposite propagating mode,

3. Results and discussions

3.1. $\mathcal{P}\mathcal{T}$ symmetry in gain-loss balanced waveguides

The most commonly used optical structures in $\mathcal{P}\mathcal{T}$ symmetry systems are gain-loss balanced waveguides. Here, we consider a simple example, see Fig. 1, to illustrate the symmetric relations of the vector field of a single mode or between two conjugated modes under the $\mathcal{P}\mathcal{T}$ symmetry operation, depending on whether the $\mathcal{P}\mathcal{T}$ symmetry breaking occurs or not. We consider elliptical waveguide core with the semi-major (semi-minor) of 1.5µm (1µm). The material in the waveguide core region is isotropic, i.e., ε_r = 4 − iτ on the left hand side, while ε_r = 4 + iτ on the right hand side. The waveguide core is embedded in air with operation wavelength 4 µm. The eigenstates and eigenvalues β of the gain-loss balanced waveguides, as well as others throughout the paper are obtained by full-wave simulations using COMSOL MULTIPHYSICS [15]. As the magnitude of gain/losses (τ) increases, $\mathcal{P}\mathcal{T}$ symmetry breaking occurs, the real parts of two eigenvalues β merger together and the overlapped imaginary part of the two eigenvalues, i.e., Im(β) = 0, bifurcates. The exact bifurcation location of β in τ is coined as the exceptional point (EP). As evident in Fig. 1, the pair of modes with n_{ef f} = β/k₀ of 1.61605 and 1.34651 (τ=0.4) evolve to the modes with n_{ef f} = 1.44146 ± 0.194268i (τ=1.2) as τ crosses EP (in-between 0.4 and 1.2). As shown in Figs. 1(a)–1(d), the field before EP remains unchanged under the subsequent $\mathcal{P}$ (r → −r) and $\mathcal{T}$ (complex conjugation) operations. While both the eigenstates and eigenvalues β of two modes after EP become conjugate complex to each other, see Figs. 1(e)–1(h), under the subsequent $\mathcal{P}$ and $\mathcal{T}$ operations. Hence, the symmetric relations of the eigenstates and eigenvalues shown Fig. 1 are consistent with the predictions by Eq. (11).

 

Fig. 1 The x and y-components of the electric field for a pair of modes supported by the gain-loss balanced waveguides before and after the exceptional point. The eigenstates before/after EP are shown in the first/second row. (a/b) The dominating electric field component, i.e., Re(E_y)/Im(E_y), for the mode with n_{nef f} = 1.61605; (c/d) the dominating electric field component, i.e., Re(E_x)/Im(E_x), for the mode with n_{nef f} = 1.34651. If the $\mathcal{P}$ operation is applied, i.e., r → −r, the real parts of the field in (a,c) remains unchanged, while the imaginary parts of the field in (b,d) change sign. The x-components of the electric field, including Re(E_x) and Im(E_x), for the two mode with conjugate n_{ef f} = 1.44146 ± 0.194268i are shown (e–h). Evidently, the field plot in (e) can be transformed to that in (g) under $\mathcal{P}$ operation. Similarly, the field plot in (f) can be transformed to that in (h) under $\mathcal{P}$ operation, but up to a sign difference.

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It is worthy to point out that the gain-loss balanced waveguide also obeys chiral symmetry, see discussion in Section 2.2. Provided one gets the eigenstate and eigenvalue β₀ of gain-loss balanced waveguides, as a consequence of chiral symmetry, −β₀ would also be the eigenvalue even after EP. And the relationship between the modes with opposite eigenvalue is just given by the chiral operation.

3.2. Parity symmetry in anisotropic waveguides

To illustrate the parity symmetry, we consider an anisotropic waveguide as shown in Fig. 2. The cross-section of the waveguide is elliptical, thus has the parity symmetry. In the waveguide, the semi-major and semi-minor axis are 1 µm and 0.6 µm, respectively. The relative permittivity is ${\overline{\mathit{\epsilon }}}_r=\left(\begin{array}{ccc}10& 0& 4i\\ 0& 10& 0\\ -4i& 0& 10\end{array}\right)$ corresponding to magneto-optical materials, and the permeability µ_r is 1, with background medium air. As predicated in Section 2.4, the transverse electric field in backward propagating mode is same as that in forward mode under the parity operation (r → −r), while the magnetic field transforms in a similar fashion but acquire an additional sign flip. Comparing Figs. 2(a) and 2(b) with Figs. 2(c) and 2(d), it’s clear that the electric field x component and magnetic field x component are consistent with the predictions from Section 2.4.

 

Fig. 2 The x-component as well as the vector field plots of the forward ((a)–(b)) and backward ((c)–(d)) propagating modes in anisotropic waveguide with ellipse-cross-section. The x-component of normalized electric/magnetic field is shown in (a,c)/(b,d), and the vector plots of the in-plane electric field (magnetic field) are also shown in (a,c)/(b,d) indicated by the arrows, the length of which is proportional to the magnitude of the vector field. In (a) and (c), the vector field plots take the real part of electric field, and the imaginary part of the magnetic field are taken in (b) and (d). Both the effective refractive index of forward and backward modes are 2.4668.

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3.3. Rotational symmetry in optical fiber

According to the dual symmetry of Maxwell equation, the two forward propagating modes Ψ₁ and Ψ₂ are degenerated provided ε_r = µ_r, which can be easily proved by exchanging the permittivity tensor and permeability tensor in the Hamiltonian. In the following, we will show that rotational symmetries in optical waveguides can protect the degeneracy of the two forward propagating modes Ψ₁ and Ψ₂ without ε_r = µ_r via concrete examples, i.e., circular optical fiber or square optical waveguides, under certain conditions.

This differences between the pure TE/TM modes and the HE/HE modes lead to the following fact: one mode in each HE/EH mode pair within the aforementioned truncated mode set in circular fiber can be transformed to the other by rotating their transverse field globally with a constant angle, such statement does not hold for pure TE/TM modes, see details in Appendix B. As an example, we pick out four modes of optical fiber as shown in Figs. 3(a)–3(d). Two observations can be seen: (1) despite the variation of the rotational angle, the TE₀₁ in Fig. 3(a) or TM₀₁ in Fig. 3(b) can only be rotated to itself [16]; (2) while the field orientations of ${\text{HE}}_{21}^e$ in Fig. 3(c) and ${\text{HE}}_{21}^o$ in Fig. 3(d) can be exchanged by rotating π/4. In consistency with the discussion in Section 2.7, the operator R_tt (θ) Ψ(R (θ)⁻¹ r) can be seen as a global rotation of the electric field orientation in Fig. 3 with angle θ. Therefore, ${\text{HE}}_{21}^e$ and ${\text{HE}}_{21}^o$ satisfy Eq. (15) with the angle $\theta =\frac{\pi }{4}$. Moreover, for any HE/EH mode pair satisfies Eq. (15) with corresponding rotating angle, i.e., $\frac{\pi }{2}$ for HE₁₁ (l = 0) and $\frac{\pi }{6}$ for HE₃₁ (l = 2), the polarization degeneracy exists. Due to the fact that the circle-core fiber has continuous rotational symmetry, any HE/EH mode pair are degenerated. As for TE₀₁ and TM₀₁, there is no such relation for any mode pair. As for a waveguide with discrete rotational symmetry, if the azimuthal quantum number (l) of modes are consistent with the discrete symmetry of waveguide core, the degeneracy emerges. Otherwise the degeneracy vanishes. For example, in square-core fiber, the HE₂₁ shown in Figs. 3(e) and 3(f) modes are not degenerate, because square is not invariant under rotating with $\frac{\pi }{4}$. However, the HE₁₁ in Figs. 3(g) and 3(h) modes in square-core fiber is also degenerate, since square cross section remains the same under rotation of $\frac{\pi }{2}$.

 

Fig. 3 (a–d) TE₀₁, TM₀₁, ${\text{HE}}_{21}^e$ and ${\text{HE}}_{21}^o$ in circular optical fibers. The four modes reduce into LP₁₁ under weakly guiding. (e–h) ${\text{HE}}_{21}^e$, ${\text{HE}}_{21}^o$, ${\text{HE}}_{11}^e$ and ${\text{HE}}_{11}^o$ in square-core waveguide. The black circle or square indicate the borders of the waveguide-core, and arrows show the electric field orientation. The color plots are the azimuthal component, i.e., E_θ, of electric field. The radius of circle-core in (a–d) is 5 µm, the relative permittivity ε_r is 4, and the background material is air. The effective refractive index is 1.99154 in (a), 1.99117 in (b), and 1.99135 in both (c) and (d). The length of square in (e–h) is 9µm, the material properties are the same as (a–d). The effective refractive index is 1.99118 in (e), 1.99093 in (f), and 1.99643 in both (g) and (h). Work wavelength is 1.55µm

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4. Conclusion

In conclusion, we provide a systematic approach to classify the symmetric properties of waveguide modes in presence of gain/losses, anisotropy/bi-anisotropy, as well as the rational symmetry in the geometric cross-section. By eliminating the longitudinal field components (e_z and h_z), we derive the waveguide Hamiltonian that fully characterizes the waveguide modes. With the proper symmetry operations, i.e., chiral/time reverse/parity symmetry, associated with waveguide Hamiltonian, one can easily build up the relations between forward and backward propagating modes. As for the $\mathcal{P}\mathcal{T}$ symmetry, we can identity the symmetric properties of mode profile if the $\mathcal{P}\mathcal{T}$ symmetry is fulfilled. For the cross-section with rotational symmetry, we study how the rotation symmetry gives rise to the polarization degeneracy, illustrated by circular fiber and waveguide with square cross-section.

In general reciprocal waveguides where the simple connection between forward and backward propagating modes is lost, our work provides a shortcut to retrieve the relation between forward and backward propagating modes, thus builds up the so-called dual mode sets that are essential in coupled mode theory in waveguides. The derived Hamiltonian as well as the symmetry operation don’t rely on any specific material parameters or chosen geometry, and can be applied to any waveguide system, as long as certain symmetry relation (not limited within the symmetries discussed in the paper) is fulfilled. Importantly, our approach can be applied to analysis waveguide modes without knowing the exact field distribution, thus simplifies the modal analysis and can be useful for wave-guiding design.