Classification of symmetry properties of waveguide modes in presence of gain/losses, anisotropy/bianisotropy, or continuous/discrete rotational symmetry

We study the symmetric properties of waveguide modes in presence of gain/losses, anisotropy/bianisotropy, or continuous/discrete rotational symmetry. We provide a comprehensive approach to identity the modal symmetry by constructing a $4\times4$ waveguide Hamiltonian and searching the symmetric operation in association with the corresponding waveguides. We classify the chiral/time reversal/parity/parity time/rotational symmetry for different waveguides, and provide the criterion for the aforementioned symmetry operations. Lastly, we provide examples to illustrate how the symmetry operations can be used to classify the modal properties from the symmetric relation between modal profiles of several different waveguides.


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 fields, 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 PT posses interesting features, i.e., real eigenvalues though the Hamiltonian being non-Hermitian, and exceptional points (EPs) where the transition of PT 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 negative propagating modes (-β) can be considered as a perfect image of the forward propagating modes (β). It is interesting to ask how the forward and backward propagating modes are related, if the waveguide materials contain gain/losses, anisotropy, or bianisotropy? One also notes that if the waveguide cross-section contains rotational symmetries, 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 waveguide mode are vector fields, 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 fields 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 operations 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.

Waveguide Hamiltonian
We take a time harmonic dependence e iωt for the electromagnetic waves throughout this paper. The source-free Maxwell's equations for general bianisotropic waveguide read as follows,  (1) can be reformulated into 4 components equation, by eliminating the e z 2d (x, y) and h z 2d (x, y) via the expressions e z 2d (x, y) = The resulted equations for the in-plane field components can be written in a compact form, where the Hamiltonian H given by x y r ∂x ∂y k 0μ z z r T is the eigenstate, which contains the in-plane field components. In Eq. (2), we limit our self to study the mode properties of the waveguide within the truncated mode set, with particular emphasis on the symmetry relations among the polarizations, as well as that between the forward propagating modes and the backward propagating modes. The truncated mode set is defined as the waveguide modes, which share the field configuration labeled by the same quantum numbers in the traverse plane. For simplicity, we investigate the waveguides with single core structure, the medium of which could be active, lossy, anisotropic or bianisotropic. The geometric cross section of the waveguide core structure could be irregular, or highly symmetric. The background is homogeneous and isotropic.
Corresponding to the 4 × 4 matrix form Hamiltonian, there will be 4 eigenmodes Ψ + 1 , Ψ + 2 , Ψ − 1 and Ψ − 2 in the truncated mode set, with eigenvalue being β + 1 , β + 2 , β − 1 and β − 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 eigenfields of each region, and then apply the boundary condition to connect the fields from different regions such that the eigenfields 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 eigenfields from different region are 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.

Chiral symmetry
We study the degeneracy between opposite propagating modes (Ψ + 1 and Ψ − 1 or Ψ + 2 and Ψ − 2 ). Here, an unitary matrix is introduced as an operator to describe a chiral transformation. As the operator σ acts on a state Ψ, it reverses the sign of transverse magnetic field while the transverse electric fields remain unchanged, and the original and transformed transverse electromagnetic fields can be seen as left-handed and right-handed systems. Since the Poynting vector P is defined as P = E × H, the chiral operation σ changes the propagation direction of power flow, thereby builds the connection between forward and backward propagating modes. If the termsε zt r ,ε tz r ,μ zt r ,μ tz r andχ in Hamiltonian H vanish, then H in Eq. (2) is reduced to, A close examination shows that the following relation for the reduced waveguide Hamiltonian H in Eq. (4) holds, which means if β 1 is the eigenvalue of H with eigenstate Ψ 1 , the −β 1 would also be the eigenvalue with eigenstate σΨ 1 . In other words, for a given forward propagating mode, there is a degenerate backward propagating mode, and the eigen-fields transform to each other by the symmetry operation σ, provided that the constraints onε r ,μ r andχ are fulfilled.

Time reversal symmetry
Next, we introduce the time reversal operator T :p → −p, i ⇒ −i, wherep is the momentum operator [6,7]. In general, this operator can be represented as T = UK, where U is a unitary matrix and K is complex conjugation [8]. 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, T = σK.
As the operator K acts on the Hamiltonian, all the i in Eq. (2) reverses sign, and all the elements in the permittivity tensorε r , permeability tensorμ r andχ in Eq. (2) take the complex conjugate. If the waveguide is invariant under time reversal operation, which requires all the these elements in material tensors, i.e.,ε r ,μ r andχ to be real numbers, we shall have, Similar to Eq. (4), the Hamiltonian also reverses sign under the time reversal operation. Therefore, as a result of Eq. (7), the forward and backward propagating modes are degenerated, but up to a sign difference in the eigenvalues (β), the eigenstates are related by operator T . In contrast to chiral symmetry operator, we don't necessarily need the reduced Hamiltonian in Eq. (4) for T operator, but the time reversal symmetry indeed requires that all the elements in the material tensors (ε r ,μ r andχ) to be real. And the transformation between the fields of the degenerate modes, not only needs σ, but also needs take the complex conjugate. Despite those differences in chiral symmetry operator and the time reversal operator, both can be applied to scenarios, in whichχ is zero,ε r ,μ r are real and without tz,zt elements, and the two symmetry operation yields exactly the same results. Same as chiral symmetry, time reversal operator T doesn't perform any action on space , thus there is no constraint on the geometry structure of waveguide.

Parity symmetry
We proceed to discuss the symmetry operation that changes the coordinates, for example, parity operator P, r → −r,p → −p, where r is the position operator and only contains transverse coordinate (x, y) [6,7]. The optical properties of waveguide are essentially determined by the spatial dependent permittivity and permeability, i.e.,ε r (r) andμ r (r). Consideringχ=0, Eq. (2) can be reformulated as: the first r in Hamiltonian H represents the coordinates that get differentiated, all the rest r in Eq. (8) simply represents the spatial dependence of material tensors and wave-function. The parity operator P also contains a unitary matrix σ and an operator that reverses coordinate. As the operator P acts on Hamiltonian in Eq. (8), one shall have the following equation PH (r,ε r (r) ,μ r (r)) P −1 = σH (−r,ε r (−r) ,μ r (−r)) σ −1 = −H (r,ε r (−r) ,μ r (−r)) . (9) If the cross-section of the waveguide is invariant under P, i.e.,ε r (−r) =ε r (r) andμ r (−r) = µ r (r), one obtains PH (r,ε r (r) ,μ r (r)) P −1 PΨ (r) = −H (r,ε r (r) ,μ r (r)) PΨ (r) = βσΨ (−r) , Consequently, PΨ (r) = σΨ (−r) is the degenerated mode (opposite propagation direction) of original state Ψ (r). In comparison with Eq. (5) in chiral symmetry, parity symmetry operation does not require that those components (ε zt r ,ε tz r ,μ zt r andμ tz r ) vanish, but reverses the coordinates of the fields before performing σ-operation. Intuitively, it can be understood that the presence of ε tz r orε zt r elements inε r orμ r breaks the chiral symmetry between the forward and backward propagating modes, while the presence of parity symmetry in the structure of cross-section restore it. Whenχ can't be ignored, the conclusion will also be kept underχ (r) = −χ (−r).

PT 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 P/T symmetry. In this subsection, we continue to discuss the the symmetric properties induced by combining the two symmetry operations together, i.e., PT symmetry, which has been examined extensively in the last few years [5-7, 9, 10]. As the operator P and T both act on Hamiltonian H, one obtains PT H (p, r, t) (PT ) −1 = H * (p, −r, −t). If the optical systems are PT symmetric (here we only concerns isotropic medium), i.e., ε r (r) = ε * r (−r), µ r (r) = µ * r (−r), one find the waveguide Hamiltonian H commutes with the PT operator, i.e., PT H (PT ) −1 = H, leading to PT H (r,ε r (r) ,μ r (r)) (PT ) −1 PT Ψ (r) = H (r,ε r (r) ,μ r (r)) Ψ * (−r) = β * Ψ * (−r) . (11) From Eq. (8) and Eq. (11), one immediately finds out the fact that if Ψ (r) is the eigenmode for Hamiltonian with eigenvalue β, its complex conjugate partner with reversed coordinates Ψ * (−r) would also be the eigenmode with eigenvalue β * . Before the PT symmetry is broken, the eigenvalues are always real number, with β * = β and Ψ * (−r) = Ψ (r). Once the PT symmetry is broken, β * and β are different values, Ψ * (−r) and Ψ (r) are separated eigenstates of H. The media can be anisotropy in time reversal symmetry or parity symmetry respectively, actually, the media under PT symmetry can also be anisotropy(See Table. (1)).

Rotation symmetry
We continue to study the degeneracy between the polarization states Ψ 1 , Ψ 2 due to the rotational symmetry of the cross-section of waveguides. Considering the structure symmetry of the crosssection can be encoded into the optical properties of the material, we use Eq. (8) that explicitly encloses the coordinate-dependent material tensors, i.e.,ε r (r) andμ r (r). Due to the symmetry requirement, we only consider isotropic waveguides such as ordinary optical fiber for simplicity. To this end, the Hamiltonian H can be reduced as, If an eigenstates Ψ 1 (r) in Eq. (8) can be rotated anticlockwise by a constant angle θ to another eigenstates Ψ 2 (r), which can be described by the following equation, where the polarization rotation operator P R (θ) = cos θ − sin θ 0 0 sin θ cos θ 0 0 0 0 cos θ − sin θ 0 0 sin θ cos θ , and the coordinate rotation operator R (θ) = cos θ − sin θ sin θ cos θ . As can be seen, the rotation of the vector field is in sharp contrast to the rotation of a scalar field: if one wants to rotate a scalar anticlockwise, one just rotates the coordinate system clockwise by same angle; as for vector field, one need to consider the rotation between the field components beside the rotation of each components, as described by Eq. (13). As a side remark, Ψ 1 and Ψ 2 can be considered as the polarization modes associated with the same field configuration, such that the in-plane vector fields of the two modes are always perpendicular, i.e., Ψ 2 = P R π 2 Ψ 1 . As the operator P R (θ) acts on the Hamiltonian H, see Eq. (12), one shall obtain, where It's interesting to note the fact that the operator P R (θ) acting on H is equivalent to rotate the differential Coordinates in H, withε r (r) andμ r (r) unchanged. With the substitution of Eq. (13) and Eq. (14) into Eq. (8), one obtains If the cross-section of waveguide is invariant under the rotation of θ, we can get H (r,ε r (r) ,μ r (r)) Ψ 2 (r) = β 1 Ψ 2 (r) from Eq. (15), thus establishes the symmetric (degenerate) relation between the two polarization modes. When the media is on longer isotropy, we can get the same conclusion with constraint thatRε r R −1 r R −1 =ε r (r),Rμ r R −1 r R −1 =μ r (r) and Rχ R −1 r R −1 =χ (r), whereR = cos θ − sin θ 0 sin θ cos θ 0 0 0 1 .

Transformation of vector field
To get a comprehensive impression of symmetry operations discussed in this paper, we list the five different symmetry operations 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 two symmetry operations establish the relationship between two modes with same propagating direction.
The transverse electromagnetic field components, which are the eigenfunction Ψ of waveguide Hamiltonian, is essentially a vector field. Considering the sharp contrast between rotating vector fields and rotating scalar fields, it is necessary to give formal expressions to describe how the vector and scalar fields are rotated. According to [11,12], 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 e t = e x , e y T ), one shall have, and where the operator R is the aforementioned coordinate rotating operator, and R the rotating operation that reshuffles different components of the vector fields. Rotating a scalar field is equivalent to rotating coordinates as described in Eq. (16). Evident from (17), there are more evolved in the rotation of a vector field. In short, we could decompose the rotation of vector field into two steps: (1) reshuffling the components of the vector field, and (2) coordinate rotation. Thus, the action of step (1) R e t (r) and step (2) e t R −1 r are very different, one acting on the field components, the other on the coordinates of each components of the vector field. We further explain the subtle difference via rotating the electric field, i.e., represented by the position-dependent arrows. The R operator in (17) acts on the electric field directly (same as σ, T and P R in Table. (1)), only changes the orientation of the arrow without moving the position of arrows, while the R operation in e t R −1 r acts on the coordinates of the arrows, only changes the arrow position without changing the orientation of the arrow. 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 step (1). In Sections 2.4, 2.5 and 2.6, those symmetry operations can be considered as combined operations of step (1) and step(2).

PT symmetry in gain-loss balanced waveguides
The most commonly used optical structures in PT symmetry systems are gain-loss balanced waveguides. Here, we consider a simple example, see Fig. 1, to illustrate the symmetric relations of the vector fields of a single mode or between two conjugated modes under the PT symmetry operation, depending on whether the PT 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 eigen-fields 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 [13]. As the magnitude of gain/losses (τ) increases, PT 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, 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 eigen-field and eigenvalue β 0 of gain-loss balanced waveguides, as a consequence of chiral symmetry, -β 0 would also be the eigenvalue even after EP. And the relationship between the modes with opposite eigenvalue is just given by the chiral operation.

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 C 2z symmetry, which is equivalent to the coordination transform r → −r in the 2D transverse plane. In the waveguide, the semi-major and semi-minor axis are 1 µm and 0.6 µm, respectively. The relative permittivity isε r = 10 0 4i 0 10 0 −4i 0 10 corresponding to magneto-optical materials, and the permeability µ r 1, with background medium air. As predicated in Section 2.4, the transverse electric fields in backward propagating mode are 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 Fig. 2 (a-b) with (c-d), it's clear that the electric field x component and magnetic field x component are consistent with the predications from Section 2.4.

Rotational symmetry in optical fiber
According to the dual symmetry of Maxwell equation, the two forward propagating modes Ψ 1 and Ψ 2 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 Ψ 1 and Ψ 2 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 fields globally with a constant angle, such statement does not hold for pure TE/TM modes, see details in Apppendix A.