## Abstract

Faraday’s and Ampere’s laws are converted to matrix operator form and rearranged such that the unknown relative permittivity and relative permeability tensors can be determined. The material and geometry of cylindrically symmetric optical resonator structures are determined through the electric and magnetic field component profiles and complex angular frequency of a proposed localized state. This differs from the usual utilization of the electromagnetic wave equations, solving for states given the material properties and geometry. Thus the technique presented here is an inverse numerical process. The theoretical expressions are provided based on a Fourier-Bessel numerical approach which is highly suitable for cylindrical geometry resonators. Without loss of the generality of the technique, examples of resonant structure determination are presented for non-magnetic and diagonal relative permittivity tensor. Axial field propagation is included to demonstrate the design capabilities related to optical fiber and photonic crystal fiber structures.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

OSA Recommended Articles Spherical space Bessel-Legendre-Fourier localized modes solver for electromagnetic waves

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Opt. Express 23(20) 25717-25737 (2015) Dielectric profile segmentation used to reduce computation effort of the Fourier-Bessel mode solving technique

Robert C. Gauthier and Seyed Hamed Jafari
Opt. Express 23(11) 14288-14300 (2015) ### References

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1. R. C. Gauthier, “Reformulation of the Fourier-Bessel steady state mode solver,” Opt. Commun. 375, 63–71 (2016).
[Crossref]
2. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[Crossref] [PubMed]
3. M. A. Alzahrani and R. C. Gauthier, “Spherical space Bessel-Legendre-Fourier localized modes solver for electromagnetic waves,” Opt. Express 23(20), 25717–25737 (2015).
[Crossref] [PubMed]
4. G. Arfken, Mathematical Methods for Physicists (Academic Press, 1970)
5. R. C. Gauthier and S. H. Jafari, “Dielectric profile segmentation used to reduce computation effort of the Fourier-Bessel mode solving technique,” Opt. Express 23(11), 14288–14300 (2015).
[Crossref] [PubMed]
6. H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79(10), 7483–7492 (1996).
[Crossref]
7. R. C. Gauthier, “Anisotropic resonator analysis using the Fourier-Bessel Mode solver,” Opt. Commun. 410, 317–327 (2018).
[Crossref]
8. T. C. Choy, Effective Medium Theory: Principles and Applications (Oxford University Press, 2016).

#### 2018 (1)

R. C. Gauthier, “Anisotropic resonator analysis using the Fourier-Bessel Mode solver,” Opt. Commun. 410, 317–327 (2018).
[Crossref]

#### 2016 (1)

R. C. Gauthier, “Reformulation of the Fourier-Bessel steady state mode solver,” Opt. Commun. 375, 63–71 (2016).
[Crossref]

#### 1996 (1)

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79(10), 7483–7492 (1996).
[Crossref]

#### 1991 (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[Crossref] [PubMed]

#### Benisty, H.

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79(10), 7483–7492 (1996).
[Crossref]

#### Gauthier, R. C.

R. C. Gauthier, “Anisotropic resonator analysis using the Fourier-Bessel Mode solver,” Opt. Commun. 410, 317–327 (2018).
[Crossref]

R. C. Gauthier, “Reformulation of the Fourier-Bessel steady state mode solver,” Opt. Commun. 375, 63–71 (2016).
[Crossref]

#### Jafari, S. H.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[Crossref] [PubMed]

#### Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[Crossref] [PubMed]

#### J. Appl. Phys. (1)

H. Benisty, “Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,” J. Appl. Phys. 79(10), 7483–7492 (1996).
[Crossref]

#### Opt. Commun. (2)

R. C. Gauthier, “Anisotropic resonator analysis using the Fourier-Bessel Mode solver,” Opt. Commun. 410, 317–327 (2018).
[Crossref]

R. C. Gauthier, “Reformulation of the Fourier-Bessel steady state mode solver,” Opt. Commun. 375, 63–71 (2016).
[Crossref]

#### Phys. Rev. B Condens. Matter (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[Crossref] [PubMed]

#### Other (2)

G. Arfken, Mathematical Methods for Physicists (Academic Press, 1970)

T. C. Choy, Effective Medium Theory: Principles and Applications (Oxford University Press, 2016).

### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

### Figures (16)

Fig. 1 Left – Top view of the Bragg Rings resonator structure. Right – Radial plot of the relative permittivity for the Bragg ring structure.
Fig. 2 TE-TM state wavelength space obtained between 1 and 2 µm plotted against arbitrary state index. Indicated$E z$and$E φ$states used in the inverse FB process to compute relative permittivity profiles needed to support selected states.
Fig. 3 Radial dependence of the relative permittivity profile (solid trace) computed through the use of the TM state (dashed trace) and angular frequency information.
Fig. 4 Radial dependence of the relative permittivity profile (solid trace) computed through the use of the TE state (dashed trace) and angular frequency information.
Fig. 5 TE and TM state wavelength dependence on axial propagation constant. TM states are horizontal dotted lines and show no axial propagation dependence, see Eq. (10). TE states are downward sloping dotted lines due to the axial propagation constant dependence of Eqs. (8) and (9).
Fig. 6 Radial dependence of the Bragg ring relative permittivity computed using the TE polarized state with axial propagation factor $k z =0.5( 2π/T )$.
Fig. 7 Left - Hexagonal array of air holes in silicon background. Center - Localized monopole state supported by the hexagonal array. Right - Computed relative permittivity obtained using the supported state (Center) and angular frequency. Relative permittivity shown in two level format to highlight hexagonal array features determined.
Fig. 8 Left - Mapped$( r,φ )→( x,y )$ $E z$field profile for the single Gaussian in the radial direction and of mode order 10 in azimuthal. This profile is proposed and the material space required to support the state is determined. Right - Grey scale amplitude profile in polar coordinates.
Fig. 9 Left - Mapped$( r,φ )→( x,y )$ $H r$field profile determined using Faraday’s law and the single Gaussian of Fig. 8. Right - Grey scale amplitude profile in polar coordinates.
Fig. 10 Left - Mapped$( r,φ )→( x,y )$ $H φ$field profile determined using Faraday’s law and the single Gaussian of Fig. 8. Right - Grey scale amplitude profile in polar coordinates.
Fig. 11 Relative permittivity (solid trace) profile obtained when the single Gaussian mode profile (dashed trace) is chosen as the design state. “Region of Interest” indicates values required to support the Gaussian state. Outside this region relative permittivity values are somewhat arbitrary as the field component is effectively zero.
Fig. 12 Masked relative permittivity profile used to confirm existence of single Gaussian mode profile as initially proposed. Profile in “Region of Interest” was obtained using inverse numerical process. Other regions set to background level of 1.0.
Fig. 13 Left - Proposed $E z$field profile for the single Gaussian. Right – Forward FB $E z$field profile obtained using the masked relative permittivity profile and displays the features of the original proposed single Gaussian state.
Fig. 14 Left - Mapped$( r,φ )→( x,y )$ $E z$field profile for the double Gaussian in the radial direction and of mode order 10 in azimuthal. This profile is proposed and the material space required to support the state is determined. Top right - Grey scale amplitude profile in polar coordinates of the proposed state. Lower right - Corresponding state returned using the masked relative permittivity profile determined using the inverse FB computation process.
Fig. 15 Relative permittivity (solid trace) profile obtained when the double Gaussian mode profile (dashed trace) is chosen as the design state. “Region of Interest” indicates values required to support the Gaussian states. Outside this region relative permittivity values are somewhat arbitrary as the field component is effectively zero.
Fig. 16 Masked relative permittivity profile used to confirm existence of double Gaussian mode profile as initially proposed. Profiles in “Region of Interest” were obtained using inverse numerical process. Other regions set to background level of 1.0.

### Equations (23)

$∇ → × E → =− ∂ B → ∂t =jω B →$
$∇ → × H → = ∂ D → ∂t =−jω D →$
$D → = ε ↔ E → = ε o ε r ↔ E → = ε o [ ε rr ε rφ ε rz ε φr ε φφ ε φz ε zr ε zφ ε zz ] E →$
$B → = μ ↔ H → = μ o μ r ↔ H → = μ o [ μ rr μ rφ μ rz μ φr μ φφ μ φz μ zr μ zφ μ zz ] H →$
$∇ → × E → = μ r ↔ ω c ℑ →$
$∇ → × ℑ → = ε r ↔ ω c E →$
$( 1 r ∂ A z ∂φ − ∂A φ ∂z )= A ∇ × ( r )= A ∇ × ( rz )+ A ∇ × ( rφ ) for r ^$
$( ∂ A r ∂z − ∂A z ∂r )= A ∇× ( φ )= A ∇× ( φr )+ A ∇× ( φz ) for φ ^$
$1 r ( ∂[ r A φ ] ∂r − ∂A r ∂φ )= A ∇× ( z )= A ∇× ( zr )+ A ∇× ( zφ ) for z ^$
$A f = ∑ qpn κ f C qpn J o ( ρ p r R ) e jqφ e j G n z e j k z z$
$A ∇ × ( r )= ∑ qpn { ( jq r ) κ z +( −j( G n + k z ) ) κ φ } C qpn J o ( ρ p r R ) e jqφ e j G n z e j k z z$
$A ∇× ( φ )= ∑ qpn { ( j( G n + k z ) J o ( ρ p r R ) κ r )+( ρ p R J 1 ( ρ p r R ) κ z ) } C qpn e jqφ e j G n z e j k z z$
$A ∇× ( z )= ∑ qpn { ( κ φ r J o ( ρ p r R ) )+( − κ φ ρ p R J 1 ( ρ p r R ) )+( −j κ r q r J o ( ρ p r R ) ) } C qpn e jqφ e j G n z e j k z z$
$∇× A → ={ [ (4) A (5) A (6) A ]= ω c [ [ B rr + B rφ + B rz ] [ B φr + B φφ + B φz ] [ B zr + B zφ + B zz ] ][ ζ r ζ φ ζ z ] }= ω c B ↔ ζ →$
$[ A → B ↔ ξ → ]⇒[ E → μ r ↔ ℑ → ] or ( [ A → B ↔ ξ → ]⇒[ ℑ → ε r ↔ E → ] )$
$[ (4) ℑ (5) ℑ (6) ℑ (4) E (5) E (6) E ]= ω c [ ε rr ε rφ ε rz 0 0 0 ε φr ε φφ ε φz 0 0 0 ε zr ε zφ ε zz 0 0 0 0 0 0 μ rr μ rφ μ rz 0 0 0 μ φr μ φφ μ φz 0 0 0 μ zr μ zφ μ zz ][ E r E φ E z ℑ r ℑ φ ℑ z ]$
$[ ∇ × ]|E= ( ω c ) 2 μ r ↔ [ ∇ × ] −1 ε r ↔ |E$
$[ ε ij μ ij ]= ∑ qpn [ κ ε ij κ μ ij ] C qpn J o ( ρ p r R ) e jqφ e j G n z$
$c ω ( 4 ) ℑ = ε rr E r + ε rφ E φ + ε rz E z c ω ( 5 ) ℑ = ε φr E r + ε φφ E φ + ε φz E z c ω ( 6 ) ℑ = ε zr E r + ε zφ E φ + ε zz E z$
$c ω ( 6 ) ℑ( r,φ,z ) = ε zz ( r,φ,z ) E z ( r,φ,z )$
$ε zz ( r,φ,z )= c ω ( 6 ) ℑ( r,φ,z ) E z ( r,φ,z )$
$ε φφ ( r,φ,z )= c ω ( 5 ) ℑ( r,φ,z ) E φ ( r,φ,z )$
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