Abstract

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

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References

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  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).
  3. D. L. Sounas and Andrea Alú, “Time-Reversal Symmetry Bounds on the Electromagnetic Response of Asymmetric Structures,” Phys. Rev. Lett. 118, 154302 (2017).
    [Crossref] [PubMed]
  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]
  6. J. Xu and Y. Chen, “General coupled mode theory in non-Hermitian waveguides,” Opt. Express 23(17), 22619–22627 (2015).
    [Crossref] [PubMed]
  7. J. Xu, B. Wu, and Y. Chen, “Elimination of polarization degeneracy in circularly symmetric bianisotropic waveguides: a decoupled case,” Opt. Express 23(9), 11566–11575(2015).
    [Crossref] [PubMed]
  8. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17): 2632–2634(2007).
    [Crossref] [PubMed]
  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).

2017 (2)

P. Chen and Y. D. Chong, “Pseudo-Hermitian Hamiltonians generating waveguide mode evolution,” Phys. Rev. A 95(6), 062113 (2017).
[Crossref]

D. L. Sounas and Andrea Alú, “Time-Reversal Symmetry Bounds on the Electromagnetic Response of Asymmetric Structures,” Phys. Rev. Lett. 118, 154302 (2017).
[Crossref] [PubMed]

2015 (2)

2014 (1)

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4(3), 031011 (2014).

2007 (1)

2002 (1)

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40: 2201 (1999).
[Crossref]

Alú, Andrea

D. L. Sounas and Andrea Alú, “Time-Reversal Symmetry Bounds on the Electromagnetic Response of Asymmetric Structures,” Phys. Rev. Lett. 118, 154302 (2017).
[Crossref] [PubMed]

Bender, C. M.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40: 2201 (1999).
[Crossref]

Bernevig, B. A.

B. A. Bernevig and T. L. Hughes, Topological insulators and topological superconductors (Princeton University, 2013), Chap. 4.
[Crossref]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40: 2201 (1999).
[Crossref]

Chen, P.

P. Chen and Y. D. Chong, “Pseudo-Hermitian Hamiltonians generating waveguide mode evolution,” Phys. Rev. A 95(6), 062113 (2017).
[Crossref]

Chen, Y.

Chong, Y. D.

P. Chen and Y. D. Chong, “Pseudo-Hermitian Hamiltonians generating waveguide mode evolution,” Phys. Rev. A 95(6), 062113 (2017).
[Crossref]

Christodoulides, D. N.

El-Ganainy, R.

Fink, Y.

Fushchich, W. I.

W. I. Fushchich and A. G. Nikitin, Symmetries of Maxwell’s Equations (D. Reidel Publishing Company, 1987).
[Crossref]

Gagnon, L.

B. Richard and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry (McGraw-Hill, 2010), Chap. 3.

Ge, L.

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4(3), 031011 (2014).

Hamermesh, M.

M. Hamermesh, Group theory and its application to physical problems (Courier Corporation, 1962) Chap. 3.

Hughes, T. L.

B. A. Bernevig and T. L. Hughes, Topological insulators and topological superconductors (Princeton University, 2013), Chap. 4.
[Crossref]

Jacobs, S. A.

Joannopoulos, J. D.

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.

Johnson, S. G.

Love, J. D.

A. W. Snyder and J. D. Love, Optical waveguide theory (Springer Science & Business Media, 2012).

Makris, K. G.

Meade, R. D.

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.

Meisinger, P. N.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40: 2201 (1999).
[Crossref]

Musslimani, Z. H.

Nikitin, A. G.

W. I. Fushchich and A. G. Nikitin, Symmetries of Maxwell’s Equations (D. Reidel Publishing Company, 1987).
[Crossref]

Richard, B.

B. Richard and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry (McGraw-Hill, 2010), Chap. 3.

Sakoda, K.

K. Sakoda, Optical Properties of Photonic crystals (Springer-Verlag, 2005).

Skorobogatiy, M.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical waveguide theory (Springer Science & Business Media, 2012).

Sounas, D. L.

D. L. Sounas and Andrea Alú, “Time-Reversal Symmetry Bounds on the Electromagnetic Response of Asymmetric Structures,” Phys. Rev. Lett. 118, 154302 (2017).
[Crossref] [PubMed]

Srednicki, M.

M. Srednicki, Quantum field theory (Cambridge University, 2007), Chap. 2,33.
[Crossref]

Stone, A. D.

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4(3), 031011 (2014).

Winn, J. N.

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.

Wu, B.

Xu, J.

J. Math. Phys. (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40: 2201 (1999).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (1)

P. Chen and Y. D. Chong, “Pseudo-Hermitian Hamiltonians generating waveguide mode evolution,” Phys. Rev. A 95(6), 062113 (2017).
[Crossref]

Phys. Rev. Lett. (1)

D. L. Sounas and Andrea Alú, “Time-Reversal Symmetry Bounds on the Electromagnetic Response of Asymmetric Structures,” Phys. Rev. Lett. 118, 154302 (2017).
[Crossref] [PubMed]

Phys. Rev. X (1)

L. Ge and A. D. Stone, “Parity-time symmetry breaking beyond one dimension: the role of degeneracy,” Phys. Rev. X 4(3), 031011 (2014).

Other (9)

M. Hamermesh, Group theory and its application to physical problems (Courier Corporation, 1962) Chap. 3.

M. Srednicki, Quantum field theory (Cambridge University, 2007), Chap. 2,33.
[Crossref]

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.

COMSOL Multiphysics 5.2: a finite element analysis, solver and simulation software. URL http://www.comsol.com/

B. Richard and L. Gagnon, Optical Waveguide Modes: Polarization, Coupling and Symmetry (McGraw-Hill, 2010), Chap. 3.

A. W. Snyder and J. D. Love, Optical waveguide theory (Springer Science & Business Media, 2012).

W. I. Fushchich and A. G. Nikitin, Symmetries of Maxwell’s Equations (D. Reidel Publishing Company, 1987).
[Crossref]

K. Sakoda, Optical Properties of Photonic crystals (Springer-Verlag, 2005).

B. A. Bernevig and T. L. Hughes, Topological insulators and topological superconductors (Princeton University, 2013), Chap. 4.
[Crossref]

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Figures (3)

Fig. 1
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(Ey)/Im(Ey), for the mode with nnef f = 1.61605; (c/d) the dominating electric field component, i.e., Re(Ex)/Im(Ex), for the mode with nnef f = 1.34651. If the 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(Ex) and Im(Ex), for the two mode with conjugate nef f = 1.44146 ± 0.194268i are shown (e–h). Evidently, the field plot in (e) can be transformed to that in (g) under P operation. Similarly, the field plot in (f) can be transformed to that in (h) under P operation, but up to a sign difference.
Fig. 2
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.
Fig. 3
Fig. 3 (a–d) TE01, TM01, HE 21 e and HE 21 o in circular optical fibers. The four modes reduce into LP11 under weakly guiding. (e–h) HE 21 e , HE 21 o , HE 11 e and 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., Eθ, of electric field. The radius of circle-core in (a–d) is 5 µm, the relative permittivity εr 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

Tables (1)

Tables Icon

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

Equations (28)

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D = ε ¯ E + χ ¯ e h H , B = μ ¯ H + χ ¯ h e E ,
H Ψ = β Ψ ,
H 1 = ( i x ε r z x ε r z z i μ r y z μ r z z y i x ε r z y ε r z z + i μ r y z μ r z z x x y k 0 ε r z z x x k 0 ε r z z i y ε r z x ε r z z + i μ r x z μ r z z y i y ε r z y ε r z z i μ r x z μ r z z x y y k 0 ε r z z y x k 0 ε r z z x y k 0 ε r z z x x k 0 ε r z z i x μ r z x μ r z z i ε r y z ε r z z y i x μ r z y μ r z z i ε r y z ε r z z x y y k 0 μ r z z y x k 0 μ r z z i y μ r z x μ r z z + i ε r x z ε r z z y i x μ r z x μ r z z i ε r x z ε r z z x ) ,
H 2 = k 0 ( i χ y x i χ y y μ r y z μ r z x μ r z z + μ r y x μ r y z μ r z y μ r z z + μ r y y i χ x x i χ x y μ r x z μ r z x μ r z z μ r x x μ r x z μ r z y μ r z z μ r x y ε r y z ε r z x ε r z z ε r y x ε r y z ε r z y ε r z z ε r y y i χ x y i χ y y ε r x z ε r z x ε r z z + ε r x x ε r x z ε r z y ε r z z + ε r x y i χ x x i χ y x ) ,
σ = ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) ,
H = ( 0 0 x y k 0 ε r z z + k 0 μ r y x x x k 0 ε r z z + k 0 μ r y y 0 0 y y k 0 ε r z z k 0 μ r x x y x k 0 ε r z z k 0 μ r x y x y k 0 μ r z z k 0 ε r y x x x k 0 μ r z z k 0 ε r y y 0 0 y y k 0 μ r z z + k 0 ε r x x y x k 0 μ r z z + k 0 ε r x y 0 0 ) .
σ = H σ 1 = H ,
T = σ K .
T H T 1 = H
H ( r , ε ¯ r ( r ) , μ ¯ r ( r ) ) Ψ ( r ) = β Ψ ( r ) ,
P H ( r , ε ¯ r ( r ) , μ ¯ r ( r ) ) P 1 = σ H ( r , ε ¯ r ( r ) , μ ¯ r ( r ) ) σ 1 = H ( r , ε ¯ r ( r ) , μ ¯ r ( r ) ) .
P H ( r , ε ¯ r ( r ) , μ ¯ r ( r ) ) P 1 P Ψ ( r ) = H ( r , ε ¯ r ( r ) , μ ¯ r ( r ) ) P Ψ ( r ) = β σ Ψ ( r ) , H ( r , ε ¯ r ( r ) , μ ¯ r ( r ) ) σ Ψ ( r ) = β σ Ψ ( r ) .
P T H ( r , ε ¯ r ( r ) , μ ¯ r ( r ) ) ( P T ) 1 P T Ψ ( r ) = H ( r , ε ¯ r ( r ) , μ ¯ r ( r ) ) Ψ * ( r ) = β * Ψ * ( r ) .
O R e x ( r ) = e x ( R 1 r ) ,
O R e t ( r ) = e t ( R 1 r ) ,
H ( r , ε r ( r ) , μ r ( r ) ) = ( 0 0 x y k 0 ε r x x k 0 ε r + k 0 μ r 0 0 x y k 0 ε r k 0 μ r y x k 0 ε r x y k 0 μ r x x k 0 μ r k 0 ε r 0 0 y y k 0 μ r + k 0 ε r y x k 0 μ r 0 0 ) .
Ψ 2 ( r ) = O R Ψ 1 ( r ) = R t t ( θ ) Ψ 1 ( R 1 ( θ ) r ) .
R t t ( θ ) H ( r , ε r ( r ) , μ r ( r ) ) R t t 1 ( θ ) = H ( R , ( θ ) r , ε r ( r ) , μ r ( r ) ) = ( 0 0 u v k 0 ε r u u k 0 ε r + k 0 μ r 0 0 v v k 0 ε r k 0 μ r v u k 0 ε r u v k 0 μ r u u k 0 μ r k 0 ε r 0 0 v v k 0 μ r + k 0 ε r v u k 0 μ r 0 0 )
O R H ( r , ε r ( r ) , μ r ( r ) ) O R   1 O R Ψ 1 ( r ) = β O R Ψ 1 ( r ) , R t t H ( R 1 r , ε r ( R 1 r ) , μ r ( R 1 r ) ) R t t 1 R t t Ψ 1 ( R 1 r ) = β 1 R t t Ψ 1 ( R 1 r ) , H ( r , ε r ( R 1 r ) , μ r ( R 1 r ) ) Ψ 2 ( r ) = β 1 Ψ 2 ( r ) .
O S e t ( r ) = S e t ( S r ) ,
σ O S H O S   1 σ 1 = H , Ψ ( r ) = σ S t t Ψ + ( S r ) ,
× E = i ω B = i ω ( μ ¯ H + χ ¯ h e E ) , × H = i ω D = i ω ( ε ¯ E + χ ¯ e h H ) ,
[ × + i k 0 χ ¯ h e r ] e 3 d ( x , y , z ) + i k 0 ( μ ¯ r ) h 3 d ( x , y , z ) = 0 , [ × i k 0 χ ¯ e h r ] h 3 d ( x , y , z ) i k 0 ( ε ¯ r ) e 3 d ( x , y , z ) = 0 ,
H ( ρ , ϕ ) = ( 0 0 ρ φ k 0 ε r ρ ρ ρ k 0 ε r ρ ρ + k 0 μ r 0 0 ϕ ρ ϕ k 0 ε r ρ k 0 μ r ϕ ρ ρ k 0 ε r ρ ρ ρ ϕ k 0 μ r ρ ρ ρ k 0 μ r ρ ρ k 0 ε r 0 0 ϕ ρ ϕ k 0 μ r ϕ + k 0 ε r ϕ ρ ρ k 0 μ r ρ ρ 0 0 ) ,
e t e / o = J m 1 ( ρ ) ( cos ( l ϕ ) sin ( l ϕ ) ) o r ( sin ( l ϕ ) cos ( l ϕ ) ) ,
e t e / o = J m 2 ( ρ ) ( sin ( l ϕ ) cos ( l ϕ ) ) o r ( cos ( l ϕ ) sin ( l ϕ ) ) .
( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) ( cos ( l ( ϕ θ ) ) sin ( l ( ϕ θ ) ) ) = ( cos ( l ϕ ( l 1 ) θ ) sin ( l ϕ ( l 1 ) θ ) ) = ( sin ( l ϕ ) cos ( l ϕ ) ) ,
( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) ( sin ( l ( ϕ θ ) ) cos ( l ( ϕ θ ) ) ) = ( sin ( l ϕ ( l + 1 ) θ ) cos ( l ϕ ( l + 1 ) θ ) ) = ( cos ( l ϕ ) sin ( l ϕ ) ) ,

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