Abstract

The Fourier-Bessel space conversion of Maxwell’s wave equations into an eigenvalue formulation is a useful numerical tool for computing the steady states of cylindrically symmetric dielectric structures. The relative dielectric profile, inverse (1/εr) present in wave equations, is split into a constant offset and corresponding spatially dependent residue and greatly reduces the matrix building time (and thus computation time) when alternate dielectric configurations are considered with identical spatial distributions. Such a process significantly speeds up the theoretical analysis of numerous optical designs, such as index of refraction sensors, hole infiltration sensors and resonator tuning. The theoretical steps involved are presented along with examples of the technique applied to the well-known Bragg resonator and central defect containing hexagonal array.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Spherical space Bessel-Legendre-Fourier localized modes solver for electromagnetic waves

Mohammed A. Alzahrani and Robert C. Gauthier
Opt. Express 23(20) 25717-25737 (2015)

Fourier–Bessel analysis of localized states and photonic bandgaps in 12-fold photonic quasi-crystals

Scott R. Newman and Robert C. Gauthier
J. Opt. Soc. Am. A 29(11) 2344-2349 (2012)

References

  • View by:
  • |
  • |
  • |

  1. R. C. Gauthier and K. Mnaymneh, “FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states,” Opt. Commun. 264(1), 78–88 (2006).
    [Crossref]
  2. Bahrampour, A. Seifalinezhad, A. Iadicicco, and A. R. Bahrampour, “Zero birefringence 8-fold photonic quasicrystal (QC) fiber,” in Proceedings of IEEE Conference on Photonics Third Mediterranean (IEEE, 2014), pp. 1–3.
  3. S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express 11(2), 167–175 (2003).
    [Crossref] [PubMed]
  4. R. C. Gauthier and M. Alzahrani, “Cylindrical space Fourier-Bessel mode solver for Maxwell’s wave equation,” Adv. Mater. 2(3), 32–35 (2013).
  5. S. R. Newman and R. C. Gauthier, “Fourier-Bessel analysis of localized states and photonic bandgaps in 12-fold photonic quasi-crystals,” J. Opt. Soc. Am. A 29(11), 2344–2349 (2012).
    [Crossref] [PubMed]
  6. J. C. Slater, Quantum Theory of Molecules and Solids McGraw-Hill, 1963, Chap. 7.
  7. G. E. Town, W. Yuan, R. McCosker, and O. Bang, “Microstructured optical fiber refractive index sensor,” Opt. Lett. 35(6), 856–858 (2010).
    [Crossref] [PubMed]
  8. I. Shavrin, S. Novotny, A. Shevchenko, and H. Ludvigsen, “Gas refractometry using a hollow-core photonic bandgap fiber in a Mach-Zehnder-type interferometer,” Appl. Phys. Lett. 100(5), 051106 (2012).
    [Crossref]
  9. W. Yuan, L. Khan, D. J. Webb, K. Kalli, H. K. Rasmussen, A. Stefani, and O. Bang, “Humidity insensitive TOPAS polymer fiber Bragg grating sensor,” Opt. Express 19(20), 19731–19739 (2011).
    [Crossref] [PubMed]
  10. H. Bao, K. Nielsen, H. K. Rasmussen, P. U. Jepsen, and O. Bang, “Fabrication and characterization of porous-core honeycomb bandgap THz fibers,” Opt. Express 20(28), 29507–29517 (2012).
    [PubMed]
  11. G. Emiliyanov, P. E. Høiby, L. H. Pedersen, and O. Bang, “Selective serial multi-antibody biosensing with TOPAS microstructured polymer optical fibers,” Sensors (Basel) 13(3), 3242–3251 (2013).
    [Crossref] [PubMed]
  12. G. Senthil Murugan, M. N. Petrovich, Y. Jung, J. S. Wilkinson, and M. N. Zervas, “Hollow-bottle optical microresonators,” Opt. Express 19(21), 20773–20784 (2011).
    [Crossref] [PubMed]
  13. M. Luo, Y. G. Liu, Z. Wang, T. Han, J. Guo, and W. Huang, “Microfluidic assistant beat-frequency interferometer based on a single-hole-infiltrated dual-mode microstructured optical fiber,” Opt. Express 22(21), 25224–25232 (2014).
    [PubMed]
  14. S. R. Newman and R. C. Gauthier, “Fourier-Bessel expansions of localized light in disorder dielectric lattices,” Proc. SPIE 8269, 82690T (2012).
    [Crossref]
  15. R. C. Gauthier, M. A. Alzahrani, and S. H. Jafari, “Theoretical examination of the slot channel waveguide configured in a cylindrically symmetric dielectric ring profile,” Opt. Commun. 329, 154–162 (2014).
    [Crossref]
  16. F. Bowman, Introduction to Bessel Functions Dover Publications, New York, 1958, pp. 17–18.
  17. K. Busch, C. T. Chan, and C. M. Soukoulis, “Techniques for band structures and defect states in photonic crystals,” Photon. Band Gap Mater. 315, 465–485 (1996).
    [Crossref]
  18. M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
    [Crossref]

2014 (2)

R. C. Gauthier, M. A. Alzahrani, and S. H. Jafari, “Theoretical examination of the slot channel waveguide configured in a cylindrically symmetric dielectric ring profile,” Opt. Commun. 329, 154–162 (2014).
[Crossref]

M. Luo, Y. G. Liu, Z. Wang, T. Han, J. Guo, and W. Huang, “Microfluidic assistant beat-frequency interferometer based on a single-hole-infiltrated dual-mode microstructured optical fiber,” Opt. Express 22(21), 25224–25232 (2014).
[PubMed]

2013 (2)

R. C. Gauthier and M. Alzahrani, “Cylindrical space Fourier-Bessel mode solver for Maxwell’s wave equation,” Adv. Mater. 2(3), 32–35 (2013).

G. Emiliyanov, P. E. Høiby, L. H. Pedersen, and O. Bang, “Selective serial multi-antibody biosensing with TOPAS microstructured polymer optical fibers,” Sensors (Basel) 13(3), 3242–3251 (2013).
[Crossref] [PubMed]

2012 (4)

S. R. Newman and R. C. Gauthier, “Fourier-Bessel expansions of localized light in disorder dielectric lattices,” Proc. SPIE 8269, 82690T (2012).
[Crossref]

I. Shavrin, S. Novotny, A. Shevchenko, and H. Ludvigsen, “Gas refractometry using a hollow-core photonic bandgap fiber in a Mach-Zehnder-type interferometer,” Appl. Phys. Lett. 100(5), 051106 (2012).
[Crossref]

S. R. Newman and R. C. Gauthier, “Fourier-Bessel analysis of localized states and photonic bandgaps in 12-fold photonic quasi-crystals,” J. Opt. Soc. Am. A 29(11), 2344–2349 (2012).
[Crossref] [PubMed]

H. Bao, K. Nielsen, H. K. Rasmussen, P. U. Jepsen, and O. Bang, “Fabrication and characterization of porous-core honeycomb bandgap THz fibers,” Opt. Express 20(28), 29507–29517 (2012).
[PubMed]

2011 (2)

2010 (1)

2009 (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

2006 (1)

R. C. Gauthier and K. Mnaymneh, “FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states,” Opt. Commun. 264(1), 78–88 (2006).
[Crossref]

2003 (1)

1996 (1)

K. Busch, C. T. Chan, and C. M. Soukoulis, “Techniques for band structures and defect states in photonic crystals,” Photon. Band Gap Mater. 315, 465–485 (1996).
[Crossref]

Albin, S.

Alzahrani, M.

R. C. Gauthier and M. Alzahrani, “Cylindrical space Fourier-Bessel mode solver for Maxwell’s wave equation,” Adv. Mater. 2(3), 32–35 (2013).

Alzahrani, M. A.

R. C. Gauthier, M. A. Alzahrani, and S. H. Jafari, “Theoretical examination of the slot channel waveguide configured in a cylindrically symmetric dielectric ring profile,” Opt. Commun. 329, 154–162 (2014).
[Crossref]

Bang, O.

Bao, H.

Busch, K.

K. Busch, C. T. Chan, and C. M. Soukoulis, “Techniques for band structures and defect states in photonic crystals,” Photon. Band Gap Mater. 315, 465–485 (1996).
[Crossref]

Chan, C. T.

K. Busch, C. T. Chan, and C. M. Soukoulis, “Techniques for band structures and defect states in photonic crystals,” Photon. Band Gap Mater. 315, 465–485 (1996).
[Crossref]

Diem, M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

Emiliyanov, G.

G. Emiliyanov, P. E. Høiby, L. H. Pedersen, and O. Bang, “Selective serial multi-antibody biosensing with TOPAS microstructured polymer optical fibers,” Sensors (Basel) 13(3), 3242–3251 (2013).
[Crossref] [PubMed]

Gauthier, R. C.

R. C. Gauthier, M. A. Alzahrani, and S. H. Jafari, “Theoretical examination of the slot channel waveguide configured in a cylindrically symmetric dielectric ring profile,” Opt. Commun. 329, 154–162 (2014).
[Crossref]

R. C. Gauthier and M. Alzahrani, “Cylindrical space Fourier-Bessel mode solver for Maxwell’s wave equation,” Adv. Mater. 2(3), 32–35 (2013).

S. R. Newman and R. C. Gauthier, “Fourier-Bessel expansions of localized light in disorder dielectric lattices,” Proc. SPIE 8269, 82690T (2012).
[Crossref]

S. R. Newman and R. C. Gauthier, “Fourier-Bessel analysis of localized states and photonic bandgaps in 12-fold photonic quasi-crystals,” J. Opt. Soc. Am. A 29(11), 2344–2349 (2012).
[Crossref] [PubMed]

R. C. Gauthier and K. Mnaymneh, “FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states,” Opt. Commun. 264(1), 78–88 (2006).
[Crossref]

Guo, J.

Guo, S.

Han, T.

Høiby, P. E.

G. Emiliyanov, P. E. Høiby, L. H. Pedersen, and O. Bang, “Selective serial multi-antibody biosensing with TOPAS microstructured polymer optical fibers,” Sensors (Basel) 13(3), 3242–3251 (2013).
[Crossref] [PubMed]

Huang, W.

Jafari, S. H.

R. C. Gauthier, M. A. Alzahrani, and S. H. Jafari, “Theoretical examination of the slot channel waveguide configured in a cylindrically symmetric dielectric ring profile,” Opt. Commun. 329, 154–162 (2014).
[Crossref]

Jepsen, P. U.

Jung, Y.

Kalli, K.

Khan, L.

Koschny, T.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

Liu, Y. G.

Ludvigsen, H.

I. Shavrin, S. Novotny, A. Shevchenko, and H. Ludvigsen, “Gas refractometry using a hollow-core photonic bandgap fiber in a Mach-Zehnder-type interferometer,” Appl. Phys. Lett. 100(5), 051106 (2012).
[Crossref]

Luo, M.

McCosker, R.

Mnaymneh, K.

R. C. Gauthier and K. Mnaymneh, “FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states,” Opt. Commun. 264(1), 78–88 (2006).
[Crossref]

Newman, S. R.

S. R. Newman and R. C. Gauthier, “Fourier-Bessel expansions of localized light in disorder dielectric lattices,” Proc. SPIE 8269, 82690T (2012).
[Crossref]

S. R. Newman and R. C. Gauthier, “Fourier-Bessel analysis of localized states and photonic bandgaps in 12-fold photonic quasi-crystals,” J. Opt. Soc. Am. A 29(11), 2344–2349 (2012).
[Crossref] [PubMed]

Nielsen, K.

Novotny, S.

I. Shavrin, S. Novotny, A. Shevchenko, and H. Ludvigsen, “Gas refractometry using a hollow-core photonic bandgap fiber in a Mach-Zehnder-type interferometer,” Appl. Phys. Lett. 100(5), 051106 (2012).
[Crossref]

Pedersen, L. H.

G. Emiliyanov, P. E. Høiby, L. H. Pedersen, and O. Bang, “Selective serial multi-antibody biosensing with TOPAS microstructured polymer optical fibers,” Sensors (Basel) 13(3), 3242–3251 (2013).
[Crossref] [PubMed]

Petrovich, M. N.

Rasmussen, H. K.

Senthil Murugan, G.

Shavrin, I.

I. Shavrin, S. Novotny, A. Shevchenko, and H. Ludvigsen, “Gas refractometry using a hollow-core photonic bandgap fiber in a Mach-Zehnder-type interferometer,” Appl. Phys. Lett. 100(5), 051106 (2012).
[Crossref]

Shevchenko, A.

I. Shavrin, S. Novotny, A. Shevchenko, and H. Ludvigsen, “Gas refractometry using a hollow-core photonic bandgap fiber in a Mach-Zehnder-type interferometer,” Appl. Phys. Lett. 100(5), 051106 (2012).
[Crossref]

Soukoulis, C. M.

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

K. Busch, C. T. Chan, and C. M. Soukoulis, “Techniques for band structures and defect states in photonic crystals,” Photon. Band Gap Mater. 315, 465–485 (1996).
[Crossref]

Stefani, A.

Town, G. E.

Wang, Z.

Webb, D. J.

Wilkinson, J. S.

Yuan, W.

Zervas, M. N.

Adv. Mater. (1)

R. C. Gauthier and M. Alzahrani, “Cylindrical space Fourier-Bessel mode solver for Maxwell’s wave equation,” Adv. Mater. 2(3), 32–35 (2013).

Appl. Phys. Lett. (1)

I. Shavrin, S. Novotny, A. Shevchenko, and H. Ludvigsen, “Gas refractometry using a hollow-core photonic bandgap fiber in a Mach-Zehnder-type interferometer,” Appl. Phys. Lett. 100(5), 051106 (2012).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

R. C. Gauthier, M. A. Alzahrani, and S. H. Jafari, “Theoretical examination of the slot channel waveguide configured in a cylindrically symmetric dielectric ring profile,” Opt. Commun. 329, 154–162 (2014).
[Crossref]

R. C. Gauthier and K. Mnaymneh, “FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states,” Opt. Commun. 264(1), 78–88 (2006).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Photon. Band Gap Mater. (1)

K. Busch, C. T. Chan, and C. M. Soukoulis, “Techniques for band structures and defect states in photonic crystals,” Photon. Band Gap Mater. 315, 465–485 (1996).
[Crossref]

Phys. Rev. B (1)

M. Diem, T. Koschny, and C. M. Soukoulis, “Wide-angle perfect absorber/thermal emitter in the terahertz regime,” Phys. Rev. B 79(3), 033101 (2009).
[Crossref]

Proc. SPIE (1)

S. R. Newman and R. C. Gauthier, “Fourier-Bessel expansions of localized light in disorder dielectric lattices,” Proc. SPIE 8269, 82690T (2012).
[Crossref]

Sensors (Basel) (1)

G. Emiliyanov, P. E. Høiby, L. H. Pedersen, and O. Bang, “Selective serial multi-antibody biosensing with TOPAS microstructured polymer optical fibers,” Sensors (Basel) 13(3), 3242–3251 (2013).
[Crossref] [PubMed]

Other (3)

F. Bowman, Introduction to Bessel Functions Dover Publications, New York, 1958, pp. 17–18.

Bahrampour, A. Seifalinezhad, A. Iadicicco, and A. R. Bahrampour, “Zero birefringence 8-fold photonic quasicrystal (QC) fiber,” in Proceedings of IEEE Conference on Photonics Third Mediterranean (IEEE, 2014), pp. 1–3.

J. C. Slater, Quantum Theory of Molecules and Solids McGraw-Hill, 1963, Chap. 7.

Cited By

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

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Top – Inverse relative dielectric constant profile for the central region of the Bragg structure (insert to Fig. 3). It can be considered as the summation of a constant offset, Ω off , and a residue Ω res =Ω Ω off as shown on the top-right. Lower left - Inverse relative dielectric constant profile in which the spatial distribution is the same as the top left trace. The lower-right indicates that the offset and residue values can be obtained by scaling the values of the top-right image using (1).
Fig. 2
Fig. 2 Left –Bragg structure consisting of three different relative dielectric constant materials. Right - The offset level is indicated by the dashed line. The central core region forms one residue spatial profile with zero values in the confining ring region. The second residue profile is formed from the wide segments of the confining rings. The second residue profile has zero values in the central core region. The eigenmatrix can be rescaled to accommodate structures having the same spatial relative dielectric distribution.
Fig. 3
Fig. 3 Plot of the residue scaling factor versus relative dielectric constant ε ' 1 .Residue rescale factor = 1 corresponds to original dielectric structure for which the matrix elements in (8) are determined. For this computation example ε ' off = ε off . Insert shows the Bragg structure with white being silicon and black the oxide.
Fig. 4
Fig. 4 Plot of the eigen-wavelengths in the 0.5 to 5 µm range versus, upper scale - matrixfile index, lower scale altered relative dielectric constant, for the 1111 eigen-matrices solved for monopole states with eigenvectors dominated by the H z field component. Double arrow used to locate the primary bandgap and single arrow points to a state localized to the central region of the dielectric structure. Two other bandgaps at lower wavelengths (higher frequencies) are observed in the figure. One of the bandgaps displays that a higher order localized monopole state is present.
Fig. 5
Fig. 5 Monopole, dipole, quadrupole and hexapole states of the primary bandgap dominated by the H z field component in the eigenvector plotted versus file index and altered relative dielectric constant. Selected mode profile plotted displaying the rotational symmetry and localized nature of the bandgap states.
Fig. 6
Fig. 6 Primary band gap range fo the silicon background hexagonal array plotted versus relative dielectric constnat within the holes of the array. Insert - Hexagonal array dielectric profile with central large defect (white – Silicon,black - air) shown in insert. Additional line in bandgap indicates the location of the monopole states (see Fig. 7 for enlarged state plot).
Fig. 7
Fig. 7 Monopole states in the 1.2 and 2.2 µm wavelength range with 75% or greater field confinement to the central disk region of the hexagonal structure. States with eigenvector dominated by the H z field component plotted. Lower axis represents the file index and altered relative dielectric constant varied from 1.0 to 2.4 in 0.1 increments (15 data files). Due to the pinching off nature of the bandgap as the dielectric contrast decreases the localized monopole is not supported for infiltration relative dielectric constant above 2.3.
Fig. 8
Fig. 8 Left – All monopole states in the 1.2 and 2.2 µm wavelength range returned from the eigen-solver dominated by the H z field component. States with 75% or greater field confinement to the central region are plotted on the right. The field confinement filter identifies and extracts localized states from all states provided by the solver. States identified with an addition O are plotted in top down column format for the intensity of the H z field component. They display the various mode symmetries composing the monopole family.
Fig. 9
Fig. 9 Top – Real part of the expansion coefficients for the 6-fold rotationally symmetric hexagonal array. Only the coefficients in the angular index range from −33 to + 33 are shown. The 6-fold rotational symmetry present forces all expansion coefficients with angular index which are not an integer multiple of the rotational symmetry to zero. Left – Positive side and absolute value of the expansion coefficients of a strong monopole (center region relative dielectric constant of value 11, wavelength 1.461 µm). Right – Positive side and absolute value expansion space for the hexapole state at wavelength 1.323 µm. Both these states show the 6-fold symmetry of the dielectric. The center figure is dominated by the zero order angular index and defines the usual monopole while the left state is dominated by the angular order of 6 of a hexapole. Inserts show field amplitude profiles.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

Ω ' off ( r,ϕ,z )=( ε off ε ' off = K off ) Ω off ( r,ϕ,z )
Ω ' res ( r,ϕ,z )=( 1 ε ' 1 1 ε ' off 1 ε 1 1 ε off = K res ) Ω res ( r,ϕ,z )
Ω= pnm κ Ω J o ( ρ p r R ) e jmϕ e j G n z
Ω= Ω off + pnm κ Ω res J o ( ρ p r R ) e jmϕ e j G n z
Ω × × E = ( ω c ) 2 E
×Ω × H = ( ω c ) 2 H
K off [ Ω off ( × × E ) ]+ K res [ Ω res ( × × E ) ]= ( ω c ) 2 E
K off [ Ω off ( × × H ) ]+ K res [ ( × Ω res × H ) ]= ( ω c ) 2 H
( E , H )= pnm κ ( E , H ) J o ( ρ p r R ) e jmϕ e j( G n )z
K off [ R r,off Φ r,off Z r,off R ϕ,off Φ ϕ,off Z ϕ,off R z,off Φ z,off Z z,off ][ R E,H Φ E,H Z E,H ]+ K res [ R r,res Φ r,res Z r,res R ϕ,res Φ ϕ,res Z ϕ,res R z,res Φ z,res Z z,res ][ R E,H Φ E,H Z E,H ]= ( ω c ) 2 [ R E,H Φ E,H Z E,H ]
K off [ R r,off Φ r,off Z r,off R ϕ,off Φ ϕ,off Z ϕ,off R z,off Φ z,off Z z,off ][ R E,H Φ E,H Z E,H ]+ i=1 n { K i res [ R i r,res Φ i r,res Z i r,res R i ϕ,res Φ i ϕ,res Z i ϕ,res R i z,res Φ i z,res Z i z,res ] } [ R E,H Φ E,H Z E,H ]= ( ω c ) 2 [ R E,H Φ E,H Z E,H ]
R r,res = 1 V [ J 1 ( ρ p r * ) ] 2 Ω κ Ω ( [ q r q Ω + q r 2 ] ) T 1 ( 0 p r , 0 p Ω , 0 p r * )+[ G n r G n Ω + G n r 2 R 2 T 1 ( 0 p r , 0 p Ω , 0 p r * ) ] 4 π 2 qn
φ r,res = 1 V [ J 1 ( ρ p r * ) ] 2 Ω κ Ω j( q φ + q Ω )[ T 1 ( 0 pφ , 0 p Ω , 0 p r * )+[ ρ pφ T o ( 0 pφ , 0 p Ω , 0 p r * ) ] ] 4 π 2 qn
Z r,res = 1 V [ J 1 ( ρ p r * ) ] 2 Ω κ Ω ( jR ρ p z )( G n Ω + G n z ) T 1 ( 1 p ϕ , 0 p Ω , 0 p r * ) 4 π 2 qn
R φ,res = 1 V [ J 1 ( ρ p φ * ) ] 2 Ω κ Ω ( j )( [ q r T 1 ( 0 p r , 0 p Ω , 0 p φ * )+ q r ρ p r T 1 ( 1 p r , 0 p Ω , 0 p φ * ) q r ρ pφ T o ( 0 p r , 1 p Ω , 0 p φ * ) ] ) 4 π 2 qn
ϕ ϕ,res = 1 V [ J 1 ( ρ p ϕ * ) ] 2 Ω κ Ω R 2 [ ( G n ϕ G n Ω + G n φ 2 + ρ p ϕ 2 R 2 ) T 1 ( 0 p φ , 0 p Ω , 0 p φ * )+ 1 R 2 T 1 ( 0 p φ , 0 p Ω , 0 p φ * )+ ρ p ϕ R 2 T o ( 0 p φ , 1 p Ω , 0 p φ * ) ρ p φ ρ p Ω R 2 T o ( 1 p φ , 1 p Ω , 0 p φ * ) ] 4 π 2 qn
Z φ,res = 1 V [ J 1 ( ρ p φ * ) ] 2 Ω κ Ω R q z ( G n Ω G n z ) T o ( 0 p z , 0 p Ω , 0 p φ * ) 4 π 2 qn
R z,res = 1 V [ J 1 ( ρ p z * ) ] 2 Ω κ Ω ( jR )[ G n r T 0 ( 0 p r , 0 p Ω , 0 p z * ) G n r ρ p r T 1 ( 1 p r , 0 p Ω , 0 p z * ) G n r ρ p Ω T 1 ( 0 p r , 1 p Ω , 0 p z * ) ] 4 π 2 qn
φ z,res = 1 V [ J 1 ( ρ p z * ) ] 2 Ω κ Ω ( R G n ϕ )( q φ + q Ω )[ T 0 ( 0 p ϕ , 0 p Ω , 0 p z * ) ] 4 π 2 qn
Z z,res = 1 V [ J 1 ( ρ p z * ) ] 2 Ω κ Ω [ ρ p z 2 T 1 ( 0 p z , 0 p Ω , 0 p z * )+( q z q Ω + q z 2 ) T 1 ( 0 p z , 0 p Ω , 0 p z * ) ρ p z ρ p z * T 1 ( 1 p z , 1 p Ω , 0 p z * ) ] 4 π 2 qn
T i ( A,B,C )= 0 1 J A ( ρ A τ ) J B ( ρ B τ ) J C ( ρ C τ ) τ i dτ

Metrics