Abstract

We present a robust and accuracy enhanced method for analyzing the propagation behavior of EM waves in z-periodic structures in (r,ϕ,z)-cylindrical co-ordinates. A cylindrical disk, characterized by the radius a and the periodicity length Lz , defines the fundamental cell in our problem. The permittivity of the dielectric inside this cell is characterized by an arbitrary, single-valued function ε(r,ϕ,z) of all three spatial co-ordinates. We consider both open and closed boundary problems. Irrespective of the type of the boundary conditions on the surface r=a, our method requires the discretization of the fields in the interior of the disk only. Inside the disk volume, we expand the fields in terms of planewaves on discrete cylindrical surfaces ri =iΔ, with Δ being the discretization step length. The fields on the nested surfaces ri =iΔ in the interior of the simulation domain are interrelated by the application of a simple, yet, powerful finite difference scheme. In free space outside the disk, the fields are expanded in terms of the closed-form eigensolutions of the Maxwell’s equations in cylindrical co-ordinates. In order to uniquely determine the involved unknown coefficients, the solutions in the interior- and exterior domains are matched on the disk’s bounding surface. Our formulation utilizes a radially-diagonalized form of Maxwell’s equations, and merely involves four (out of the six) field components. It is demonstrated that our formulation is perfectly suited, but by no means limited, to cylindrical symmetric problems.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. Joannopoulos J. D., Meade R. D., Winn J. N., �??�??Photonic Crystals: Molding the Flow of Light,�??�?? Princeton, September 1995
  2. Knight J. C., Birks T. A., Russell P. St. J., Atkin D. M., �??�??All-silica single-mode optical fiber with photonic crystal cladding,�??�?? Opt. Lett. 21, 1547-1549 (1996).
    [CrossRef] [PubMed]
  3. Cucinotta A., Selleri A., Vincetti L., Zoboli M., �??�??Holey Fiber Analysis Through the Finite-Element Method,�??�?? IEEE Photon. Technol. Lett. 3, 147-149 (1991).
  4. Chan, C.T, Yu Q. L., Ho K. M., �??�??Order-N spectral method for electromagnetic waves,�??�?? Phys. Rev. B 51, 16635- 16642 (1995).
    [CrossRef]
  5. Li Z.-Y., Lin L.-L., �??�??Photonic band structures solved by a plane-wave-based transfer-matrix method,�??�?? Phys. Rev. E 67, 046607 (2003).
    [CrossRef]
  6. Merle Elson J., Tran P., �??�??Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique, �??�?? J. Opt. Soc. Am. A 12, 1765-1771 (1995).
    [CrossRef]
  7. Li L., �??�??Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,�??�?? J. Opt. Soc. Am. A, 10, 2581-2591 (1993).
    [CrossRef]
  8. Johnson S. G., Joannopoulos J.D., �??�??Block-iterative frequency-domain methods for Maxwell�??s equations in a planewave basis,�??�?? Opt. Express 8, 173-190 (2001)<a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a>
    [CrossRef] [PubMed]
  9. </a>Bienstman P., Rigorous and Efficient Modelling of Wavelength Scale Photonic Components, doctoral dissertation 2001, Ghent University, Belgium. Available online at <a href="http://photonics.intec.rug.ac.be/download/phd 104.pdf">http://photonics.intec.rug.ac.be/download/phd 104.pdf</a>
  10. Li Z.-Y., Ho K.-M., �??�??Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,�??�?? Phys. Rev. B. 67, 165104 (2003).
    [CrossRef]
  11. Varis K., Baghai-Wadji A. R., �??�??Hybrid Planewave/Finite Difference Transfer Method for Solving Photonic Crystals in Finite Thickness Slabs,�??�?? EDMO, November 15-16, 2001, Vienna, Austria, pp. 161-166
  12. Varis K., Baghai-Wadhi A.R., �??�??Z-diagonalized Planewave/FD Approach for Analyzing TE Polarized Waves in 2D Photonic Crystals,�??�?? ACES, March 24-28, 2003, Monterey, CA, USA
  13. Varis K., Baghai-Wadji A. R., �??�??A Novel 2D Pseudo-Spectral Approach of Photonic Crystal Slabs,�??�?? ACES Special Issue, (submitted).
  14. Varis K., Baghai-Wadji A.R., �??�??A Novel 3D Pseudo-spectral Analysis of Photonic Crystal Slabs,�??�?? ACES Special Issue, (submitted).
  15. Liu S., Li L. W., Leong M. S., Yeo T. S., �??�??Theory of Gyroelectric Waveguides,�??�?? PIER, 29, 231-259 (2000).
    [CrossRef]
  16. Freund R. W., �??�??A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems,�??�?? SIAM Journal of Scientific Computing, 14, 470-482 (1993).
    [CrossRef]
  17. Nachtigal N., Freund R.W., Reeb J. C., �??�??QMRPACK user�??s guide,�??�?? ORNL Technical Report ORNL/TM-12807, August 1994, also available online at <a href="http://www.cs.utk.edu/�?santa/homepage/12807.ps.gz">http://www.cs.utk.edu/�?santa/homepage/12807.ps.gz</a>
  18. Press W. H., Teukolsky S. A., Wetterling W. T., Flannery B. P., Numerical recipes in C: the Art of Scientific Computing, Cambridge University Press, Cambridge, pp. 493-495

ACES (3)

Varis K., Baghai-Wadhi A.R., �??�??Z-diagonalized Planewave/FD Approach for Analyzing TE Polarized Waves in 2D Photonic Crystals,�??�?? ACES, March 24-28, 2003, Monterey, CA, USA

Varis K., Baghai-Wadji A. R., �??�??A Novel 2D Pseudo-Spectral Approach of Photonic Crystal Slabs,�??�?? ACES Special Issue, (submitted).

Varis K., Baghai-Wadji A.R., �??�??A Novel 3D Pseudo-spectral Analysis of Photonic Crystal Slabs,�??�?? ACES Special Issue, (submitted).

IEEE Photon. Technol. Lett. (1)

Cucinotta A., Selleri A., Vincetti L., Zoboli M., �??�??Holey Fiber Analysis Through the Finite-Element Method,�??�?? IEEE Photon. Technol. Lett. 3, 147-149 (1991).

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Johnson S. G., Joannopoulos J.D., �??�??Block-iterative frequency-domain methods for Maxwell�??s equations in a planewave basis,�??�?? Opt. Express 8, 173-190 (2001)<a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a>
[CrossRef] [PubMed]

Opt. Lett. (1)

ORNL Technical Report ORNL/TM-12807 (1)

Nachtigal N., Freund R.W., Reeb J. C., �??�??QMRPACK user�??s guide,�??�?? ORNL Technical Report ORNL/TM-12807, August 1994, also available online at <a href="http://www.cs.utk.edu/�?santa/homepage/12807.ps.gz">http://www.cs.utk.edu/�?santa/homepage/12807.ps.gz</a>

Phys. Rev. B (1)

Chan, C.T, Yu Q. L., Ho K. M., �??�??Order-N spectral method for electromagnetic waves,�??�?? Phys. Rev. B 51, 16635- 16642 (1995).
[CrossRef]

Phys. Rev. B. (1)

Li Z.-Y., Ho K.-M., �??�??Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,�??�?? Phys. Rev. B. 67, 165104 (2003).
[CrossRef]

Phys. Rev. E (1)

Li Z.-Y., Lin L.-L., �??�??Photonic band structures solved by a plane-wave-based transfer-matrix method,�??�?? Phys. Rev. E 67, 046607 (2003).
[CrossRef]

PIER (1)

Liu S., Li L. W., Leong M. S., Yeo T. S., �??�??Theory of Gyroelectric Waveguides,�??�?? PIER, 29, 231-259 (2000).
[CrossRef]

SIAM Journal of Scientific Computing (1)

Freund R. W., �??�??A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems,�??�?? SIAM Journal of Scientific Computing, 14, 470-482 (1993).
[CrossRef]

Other (4)

Varis K., Baghai-Wadji A. R., �??�??Hybrid Planewave/Finite Difference Transfer Method for Solving Photonic Crystals in Finite Thickness Slabs,�??�?? EDMO, November 15-16, 2001, Vienna, Austria, pp. 161-166

Press W. H., Teukolsky S. A., Wetterling W. T., Flannery B. P., Numerical recipes in C: the Art of Scientific Computing, Cambridge University Press, Cambridge, pp. 493-495

Joannopoulos J. D., Meade R. D., Winn J. N., �??�??Photonic Crystals: Molding the Flow of Light,�??�?? Princeton, September 1995

</a>Bienstman P., Rigorous and Efficient Modelling of Wavelength Scale Photonic Components, doctoral dissertation 2001, Ghent University, Belgium. Available online at <a href="http://photonics.intec.rug.ac.be/download/phd 104.pdf">http://photonics.intec.rug.ac.be/download/phd 104.pdf</a>

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 (4)

Fig. 1.
Fig. 1.

The geometry of test case 1. Image shows one unit cell, which is periodically replicated in the z-direction.

Fig. 2.
Fig. 2.

Dispersion diagram for the four guided modes in the geometry shown in Fig. 1. Curves with the marker “o” are computed with our method while curves with the marker “x” are obtained using the planewave method. Inset shows a zoom of the bands near the Brillouin zone edge.

Fig. 3.
Fig. 3.

The converge of the eigenmode propagation constant as a function of discrete basis size. The curve with circular tags is computed with our method, while the curve with cross tags is obtained with the eigenmode expansion method. For our method, the abscissa denotes the discretization step in the radial direction and also the number of planewave components in the z-direction, which are both equal here. For the eigenmode expansion method the horizontal axis denotes the number of eigenmodes used.

Fig. 4.
Fig. 4.

The real part of the z-directional eigenmode field pattern in a fiber with circular air holes.

Equations (42)

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

( ϕ , z , μ , ε , ω ) [ h ϕ h z e ϕ e z ] = r [ h ϕ h z e ϕ e z ]
ψ = r ψ
( 2 ) ψ = r ψ
( 2 ) ψ = 2 r 2 ψ .
( n ) ψ = n r n ψ
{ n ψ ( r , ϕ , z ) r n } | r = r 0 = ( n ) ( ϕ , z ) ψ ( r 0 , ϕ , z ) .
ψ ( r 0 + h , ϕ , z ) = n = 0 h n n ! { ( n ) ( ϕ , z ) ψ ( r 0 , ϕ , z ) } .
× E = j ω μ H
× H = j ω ε H
× f = 1 r u r r u ϕ u z r ϕ z f r r f ϕ f z ,
r f ϕ = f ˜ ϕ
1 r [ ϕ e z z e ˜ ϕ r r e z + r z e r r e ˜ ϕ ϕ e r ] = j ω μ [ h r 1 r h ˜ ϕ h z ] .
e r = 1 jωεr ϕ h z + 1 jωεr z h ˜ ϕ
z 1 jωεr ϕ h z + z 1 jωεr z h ˜ ϕ jωμ r h ˜ ϕ = r e z
jωμr h z ϕ 1 jωεr ϕ h z + ϕ 1 jωεr z h ˜ ϕ = r e ˜ ϕ
[ 0 0 A 11 A 12 0 0 A 21 A 22 B 11 B 12 0 0 B 21 B 22 0 0 ] [ e z e ˜ ϕ h z h ˜ ϕ ] = r [ e z e ˜ ϕ h z h ˜ ϕ ]
A 11 = z 1 jωεr ϕ
A 12 = z 1 jωεr z jωμ r
A 21 = ϕ 1 jωεr ϕ + jωμr
A 22 = ϕ 1 jωεr z
B 11 = ϕ 1 jωεr z
B 12 = 1 jωμr 2 z ϕ + jωε r
B 21 = 1 jωμr 2 ϕ 2 jωεr
B 22 = 1 jωμr 2 ϕ z
[ e r h r ] = [ 0 1 jωμr ϕ 0 1 jωμr z 1 jωεr ϕ 0 1 jωεr z 0 ] T [ e z e ˜ ϕ h z h ˜ ϕ ] .
f ( r , ϕ , z ) = m , n f m , n ( r ) e j k m z e j n ϕ .
Δ [ A 11 i A 12 i A 21 i A 22 i ] [ h ˜ ϕ i h z i ] + [ e ˜ ϕ i 1 2 e z i 1 2 ] = [ e ˜ ϕ i + 1 2 e z i + 1 2 ]
Δ [ B 11 i + 1 2 B 12 i + 1 2 B 21 i + 1 2 B 22 i + 1 2 ] [ e ˜ ϕ i + 1 2 e z i + 1 2 ] + [ h ˜ ϕ i h z i ] = [ h ˜ ϕ i + 1 h z i + 1 ]
h ˜ ϕ 0 0 .
l E · d l = jωμ S H · d S
E · d l = e ˜ ϕ d ϕ .
H · d S = h z d S .
jωμ S H · d S = jωμ π Δ 2 4 h z 0 .
0 2 π m , n e ˜ ϕ , m , n 1 2 e j k m z e jnϕ = jωμ π Δ 2 4 m , n h z , m , n 1 2 e j k m z e jnϕ .
m , n e ˜ ϕ , m , n 1 2 2 π δ [ n ] e j k m z .
m , n e ˜ ϕ , m , n 1 2 δ [ n ] δ [ n ̂ ] 4 π 2 e j k m z = jωμ π Δ 2 4 m , n h 0 , m , n 1 2 2 π δ [ n n ̂ ] e j k m z .
h z , m ̂ , n ̂ 0 δ [ n ̂ ] 8 jωμ Δ 2 e ϕ , m ̂ , 0 1 2 = 0 .
Φ m , n 1 = [ e ˜ ϕ e z h ˜ ϕ h z ] = [ j n H n ( λ m r ) λ 2 j k m H n ( λ m r ) rωε k m r H n ( λ m r ) 0 ] ,
Φ m , n 2 = [ e ˜ ϕ e z h ˜ ϕ h z ] = [ j rωμ k m r H n ( λ m r ) 0 n H n ( λ m r ) λ 2 k m H n ( λ m r ) ] .
Ψ ( r , ϕ , z ) = m , n [ a m , n Φ m , n 1 ( r ) + b m , n Φ m , n 2 ( r ) ] e j k m z e jnϕ .
M ( ω , K z ) f = 0 .
M ( ω , K z ) f = ρ ( ω , K z )

Metrics