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 × 4 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.
© 2017 Optical Society of America
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 . Those symmetries on one hand could be used to simplify our understanding of mode hybridization associated with complicated optical structures , on the other hand impose certain constraints to electromagnetic response . One also notes that certain optical structures based on combined symmetries of parity and time reversal posses interesting features, i.e., real eigenvalues though the Hamiltonian being non-Hermitian, and exceptional points (EPs) where the transition of 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) , 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.
2.1. Waveguide Hamiltonian
In general bianisotropic waveguide, the constitutive relation can be given as follows :Appendix A. In Eq. (2), we limit ourself 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 Hamiltonian in 4 × 4 matrix form, there will be 4 eigenmodes , , and in the truncated mode set, with eigenvalue being , , and 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 ( and or and ). Here, an unitary matrixEq. (2) is reduced to, Eq. (4) holds,
2.3. Time reversal symmetry
Next, we introduce the time reversal operator , i ⇒ −i, where is the momentum operator [8, 9]. In general, this operator can be represented as , where U is a unitary matrix and K is complex conjugation . 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,Eq. (2) flip sign, and all the elements in the permittivity tensor , permeability tensor and 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., , and to be real numbers, we shall have, Eq. (4), the sign of Hamiltonian is also flipped 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 eigenfield is related by operator . 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 (, and ) to be real. And the transformation between the field 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 which is zero, , are real and without tz,zt elements, and the two symmetry operations yield exactly the same results. Same as chiral symmetry, time reversal operator doesn’t perform any action on coordinates, thus there is no constraint on the geometry structure of waveguide.
2.4. Parity symmetry
We proceed to discuss the symmetry operation that changes the coordinates, for example, parity operator , r → −r, , 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., and. Considering , Eq. (2) can be reformulated as,Eq. (8) simply represents the spatial dependence of material tensors and wave-function. The parity operator also contains a unitary matrix σ and an operator that reverses coordinate. As the operator acts on Hamiltonian in Eq. (8), one shall have the following equation, Eq. (5) in chiral symmetry, parity symmetry operation does not require that those components (,, and ) vanish, but reverses the coordinates of the field before performing σ-operation. Intuitively, it can be understood that the presence of or elements in or breaks the chiral symmetry between the forward and backward propagating modes, while the presence of parity symmetry in the structure of cross-section restores it. When can’t be ignored, the conclusion will still be valid provided .
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 symmetry. In this subsection, we continue to discuss the the symmetric properties induced by combining the two symmetry operations together, i.e., symmetry, which has been examined extensively in the last few years [5, 6, 8, 9, 11]. As the operator and both act on Hamiltonian H, one obtains . If the optical systems are symmetric (here we only concerns isotropic medium), i.e., , , one find the waveguide Hamiltonian H commutes with the operator, i.e., , leading to,Eqs. (8) and (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 symmetry is broken, the eigenvalues are always real number, with β∗ = β and Ψ∗(−r) = Ψ(r). Once the symmetry is broken, β∗ and β are different values, Ψ∗(−r) and Ψ(r) are separated eigenstates of H. For completeness condition, the symmetry for general bianiostropic waveguide is shown in Table. 1.
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 OR acts on a scalar field (for example, x component of electric field ex) and a vector field (for example, transverse electric field ), one shall have,Eq. (12). Evident from Eq. (13), 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) and step (2) 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 operator in Eq. (13) acts on the electric field directly (same as σ and ), only changes the orientation of the arrow without moving the position of arrows, while the R operation in acts on the coordinates of the arrows, only changes the arrow position without changing the orientation of the arrow. Notably, the operator can be understood as the polarization rotation operator and is just equal to R for et in Eq. (13). 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 Eqs. (12) and (13). As for the rotation of our 4 components eigenstate Ψ, the shall be modified as .
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., and . For simplicity, we only consider isotropic waveguides such as ordinary optical fiber. To this end, the Hamiltonian H can be reduced as,Eq. (14), one shall obtain, Eq. (17), thus establishes the symmetric (degenerate relation between the two polarization modes. As the media is on longer isotropic, we can arrive at the same conclusion with constraints given by , and , where .
In analogy with the rotating operator OR, the mirror reflection operator OS  reflects a vector field by using reflection operation that acts on the coordinate and polarization,Table. 1, where .
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 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. symmetry in gain-loss balanced waveguides
The most commonly used optical structures in 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 symmetry operation, depending on whether the 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., εr = 4 − iτ on the left hand side, while εr = 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 . As the magnitude of gain/losses (τ) increases, 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 nef f = β/k0 of 1.61605 and 1.34651 (τ=0.4) evolve to the modes with nef 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 (r → −r) and (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 and operations. Hence, the symmetric relations of the eigenstates and eigenvalues shown Fig. 1 are consistent with the predictions by Eq. (11).
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 corresponding to magneto-optical materials, and the permeability µr 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.
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 εr = µr, 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 εr = µr 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 ; (2) while the field orientations of in Fig. 3(c) and in Fig. 3(d) can be exchanged by rotating π/4. In consistency with the discussion in Section 2.7, the operator Rtt (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, and satisfy Eq. (15) with the angle . Moreover, for any HE/EH mode pair satisfies Eq. (15) with corresponding rotating angle, i.e., for HE11 (l = 0) and 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 . 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 .
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 (ez and hz), 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 symmetry, we can identity the symmetric properties of mode profile if the 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.
Appendix A: The Maxwell’s equations and waveguide Hamiltonian for general bianisotropic waveguides
For monochromatic wave with angle frequency ω, we get the following equation by plugging the constitutive relation Eq. (1) into in the source-free Maxwell’s equations,Eq. (21) can further be reformulated into 4 components equation by eliminating the longitudinal terms and via the expressions and , leading the waveguide Hamiltonian given in Eq. (2).
Appendix B: Rotational symmetry of vector field in circular fiber
In circular fiber with strong guidance, the waveguide modes are described by TE0m (transverse electric modes), TM0m (transverse magnetic modes), and and (the last two are hybrid modes), depending on the existence and weighting of Ez and Hz. The indices ls represent the azimuthal quantum numbers, and ms radial quantum numbers. The pure TE and TM modes are special in the sense that all the field components have no azimuthal dependence in cylindrical coordinate (ρ, ϕ, z), such as TE01 in Fig. 3 (a). We can reformulate Hamiltonian in polar coordinate,
Due the continuous rotational symmetry, all field in optical fiber can be conveniently expressed as a ϕ-function multiplied by a ϕ-function, where ρ and ϕ are the radial and azimuthal variables . For a given TE/TM, or EH/HE mode labeled by the quantum number (m, l), the transverse electric (magnetic) field can be written asEq. (23) represents EH(l−1)m modes when l > 1 or TE0m and TM0m modes when l = 1, while Eq. (24) for HE(l+1)m modes. If l > 1 and , we can have the field in Eq. (23) under the rotation by Eq. (15) given by Eq. (24), we have,
National Natural Science Foundation of China (Grant No. 61405067); Fundamental Research Funds for the Central Universities (HUST: 2017KFYXJJ027).
The authors thank Prof. Min Yan and Dr. Meng Xiao for fruitful discussions.
References and links
1. W. I. Fushchich and A. G. Nikitin, Symmetries of Maxwell’s Equations (D. Reidel Publishing Company, 1987). [CrossRef]
2. K. Sakoda, Optical Properties of Photonic crystals (Springer-Verlag, 2005).
4. M. Skorobogatiy, S. A. Jacobs, S. G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10(21), 1227–1243 (2002). [CrossRef] [PubMed]
5. P. Chen and Y. D. Chong, “Pseudo-Hermitian Hamiltonians generating waveguide mode evolution,” Phys. Rev. A 95(6), 062113 (2017). [CrossRef]
9. C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40: 2201 (1999). [CrossRef]
10. B. A. Bernevig and T. L. Hughes, Topological insulators and topological superconductors (Princeton University, 2013), Chap. 4. [CrossRef]
11. L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4(3), 031011 (2014).
12. M. Hamermesh, Group theory and its application to physical problems (Courier Corporation, 1962) Chap. 3.
13. M. Srednicki, Quantum field theory (Cambridge University, 2007), Chap. 2,33. [CrossRef]
14. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals: molding the flow of light (Princeton university, 2011), Chap. 3.
15. COMSOL Multiphysics 5.2: a finite element analysis, solver and simulation software. URL http://www.comsol.com/
16. B. Richard and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry (McGraw-Hill, 2010), Chap. 3.
17. A. W. Snyder and J. D. Love, Optical waveguide theory (Springer Science & Business Media, 2012).