Abstract

In this article we present a novel approach for determining the electromagnetic modes of photonic multilayer structures. We combine the plane wave expansion method with the method of lines resulting in a fast and accurate computational technique which we named the plane wave admittance method. In addition, we incorporate perfectly matched layers at the boundaries parallel to the multilayer surfaces which allow for easy determination of leaky modes. The convergence of the method is verified for the case of photonic crystal slab showing very good agreement with the results obtained with full three-dimensional plane wave expansion method while the numerical effort is largely reduced. The numerical implementation of the method will be soon available on the web.

© 2005 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, a J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin 2001).
  3. N. Yokouchi, A. J. Danner, and K. D. Choquette, �??Two-Dimensional Photonic Crystal Confined Vertical-Cavity Surface-Emitting Lasers,�?? IEEE J. Sel. Top. Quantum Electron. 9, 1439�??1445 (2003).
    [CrossRef]
  4. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artec House Inc., Boston, 1995).
  5. S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, �??Photonic band gaps in periodic dielectric structures: The scalar-wave approximation,�?? Phys. Rev. B 46, 10650�??10656 (1992).
    [CrossRef]
  6. S. Johnson and J. D. Joannopoulous, �??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]
  7. S. Guo and S. Albin, �??Simple plane wave implementation for photonic crystal calculations,�?? Opt. Express 11, 167�??175 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167</a>.
    [CrossRef] [PubMed]
  8. H. S. Sözuer and J. W. Haus, �??Photonic bands: convergence problems with the plane-wave method,�?? Phys. Rev. B 45, 13962�??13972 (1992).
    [CrossRef]
  9. Ch. Sauvan, Ph. Lalanne, and J. P. Hugonin, �??Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory,�?? Opt. Quantum Electron. 36, 271�??284, 2004.
    [CrossRef]
  10. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulous, and O. L. Alerhand, �??Accurate theoretical analysis of photonic band-gap materials,�?? Phys. Rev. B 48, 8434�??8437 (1993).
    [CrossRef]
  11. A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, �??Full-vector analysis of a realistic photonic crystal fiber,�?? Opt. Lett. 24, 276�??278 (1999).
    [CrossRef]
  12. J. M. Pottage, D. Bird, T. D. Hedley, J. C. Knight, T. A. Birks, P. S. J Russell, and P. J. Roberts, �??Robust photonic band gaps for hollow core guidance in PCF made from high index glass,�?? Opt. Express 11, 2854�??2861 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2854">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2854</a>.
    [CrossRef] [PubMed]
  13. S. Shi, C. Chen, and D.W. Prather, �??Revised plane wave method for dispersive material and its application to band structure calculations of photonic crystal slabs,�?? Appl. Phys. Lett. 86, 043104 (2005), <a href= "http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=VIRT01000011000004000041000001">http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=VIRT01000011000004000041000001</a>.
    [CrossRef]
  14. P. Lalanne, �??Electromagnetic Analysis of Photonic Crystal Waveguides Operating Above the Light Cone,�?? IEEE J. Quantum Electron. 38, 800-804 (2002).
    [CrossRef]
  15. M. Qiu, �??Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,�?? Appl. Phys. Lett. 81 1163�??1165 (2002).
    [CrossRef]
  16. P. Bienstman, �??Two-stage mode finder for waveguides with a 2D cross-section,�?? Opt. Quantum Electron. 36, 5�??14, 2004.
    [CrossRef]
  17. K. Ohtaka, J. Inoue, and S. Yamaguti, �??Derivation of the density of states of leaky photonic bands,�?? Phys. Rev. B 70 035109 (2004).
    [CrossRef]
  18. S. F. Helfert, R. Pregla,�??Efficient Analysis of Periodic Structures,�?? J. Lightwave Technol. 16, 1694�??1702 (1998).
    [CrossRef]
  19. O. Conradi, S. F. Helfert, and R. Pregla, �??Comprehensive Modeling of Vertical-Cavity Laser-Diodes by the Method of Lines,�?? IEEE J. Quantum Electron. 37, 928�??935 (2001).
    [CrossRef]
  20. S. F. Helfert, A. Barcz, and R. Pregla, �??Three-dimensional vectorial analysis of waveguide structures with the method of lines,�?? Opt. Quantum Electron. 35, 381�??394 (2003).
    [CrossRef]
  21. J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185�??200 (1994).
    [CrossRef]
  22. H. Derudder, F. Olyslager, D. De Zuter, S. Van den Berghe, �??Efficient Mode-Matcing Analysis of Discontinuities in Finit Planar Substrates Using Perfectly Matched Layers,�?? IEEE Trans. Antennas and Propagation 49, 185�??195 (2001).
    [CrossRef]
  23. T. Czyszanowski, �??Comparative Analysis of Validity Limits of Scalar and Vector Approaches to Optical Fields in Diode Lasers,�?? Ph.D. diss., Technical University of Lodz (2004).
  24. S. Shi, C. Chen, and D. W. Prather, �??Plane-wave expansion method for calculating band structure of photonic crystal slabs with perfectly matched layers,�?? J. Opt. Soc. Am. A 21, 1769�??1775 (2004)
    [CrossRef]
  25. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulous, and L. A. Kolodziejski, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B 60, 5751�??5758 (1999)
    [CrossRef]

Appl. Phys. Lett. (2)

S. Shi, C. Chen, and D.W. Prather, �??Revised plane wave method for dispersive material and its application to band structure calculations of photonic crystal slabs,�?? Appl. Phys. Lett. 86, 043104 (2005), <a href= "http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=VIRT01000011000004000041000001">http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=VIRT01000011000004000041000001</a>.
[CrossRef]

M. Qiu, �??Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,�?? Appl. Phys. Lett. 81 1163�??1165 (2002).
[CrossRef]

IEEE J. Quantum Electron. (2)

P. Lalanne, �??Electromagnetic Analysis of Photonic Crystal Waveguides Operating Above the Light Cone,�?? IEEE J. Quantum Electron. 38, 800-804 (2002).
[CrossRef]

O. Conradi, S. F. Helfert, and R. Pregla, �??Comprehensive Modeling of Vertical-Cavity Laser-Diodes by the Method of Lines,�?? IEEE J. Quantum Electron. 37, 928�??935 (2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

N. Yokouchi, A. J. Danner, and K. D. Choquette, �??Two-Dimensional Photonic Crystal Confined Vertical-Cavity Surface-Emitting Lasers,�?? IEEE J. Sel. Top. Quantum Electron. 9, 1439�??1445 (2003).
[CrossRef]

IEEE Trans. Antennas and Propagation (1)

H. Derudder, F. Olyslager, D. De Zuter, S. Van den Berghe, �??Efficient Mode-Matcing Analysis of Discontinuities in Finit Planar Substrates Using Perfectly Matched Layers,�?? IEEE Trans. Antennas and Propagation 49, 185�??195 (2001).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185�??200 (1994).
[CrossRef]

J. Lightwave Technol. (1)

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

Opt. Express (3)

Opt. Lett. (1)

Opt. Quantum Electron. (3)

P. Bienstman, �??Two-stage mode finder for waveguides with a 2D cross-section,�?? Opt. Quantum Electron. 36, 5�??14, 2004.
[CrossRef]

Ch. Sauvan, Ph. Lalanne, and J. P. Hugonin, �??Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory,�?? Opt. Quantum Electron. 36, 271�??284, 2004.
[CrossRef]

S. F. Helfert, A. Barcz, and R. Pregla, �??Three-dimensional vectorial analysis of waveguide structures with the method of lines,�?? Opt. Quantum Electron. 35, 381�??394 (2003).
[CrossRef]

Phys. Rev. B (5)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulous, and L. A. Kolodziejski, �??Guided modes in photonic crystal slabs,�?? Phys. Rev. B 60, 5751�??5758 (1999)
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulous, and O. L. Alerhand, �??Accurate theoretical analysis of photonic band-gap materials,�?? Phys. Rev. B 48, 8434�??8437 (1993).
[CrossRef]

H. S. Sözuer and J. W. Haus, �??Photonic bands: convergence problems with the plane-wave method,�?? Phys. Rev. B 45, 13962�??13972 (1992).
[CrossRef]

K. Ohtaka, J. Inoue, and S. Yamaguti, �??Derivation of the density of states of leaky photonic bands,�?? Phys. Rev. B 70 035109 (2004).
[CrossRef]

S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, �??Photonic band gaps in periodic dielectric structures: The scalar-wave approximation,�?? Phys. Rev. B 46, 10650�??10656 (1992).
[CrossRef]

Other (4)

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artec House Inc., Boston, 1995).

J. D. Joannopoulos, R. D. Meade, a J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin 2001).

T. Czyszanowski, �??Comparative Analysis of Validity Limits of Scalar and Vector Approaches to Optical Fields in Diode Lasers,�?? Ph.D. diss., Technical University of Lodz (2004).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Photonic crystal slab with air layer and PML and (b) PWE unit cell.

Fig. 2.
Fig. 2.

The F(k 0) (Eq. 41) as a function of normalized frequency. The eigenmodes obtained in ref. [24] are shown with dotted lines.

Fig. 3.
Fig. 3.

(a) Light intensity and the electric field transverse distribution at the interface z=0 for k 0=0.087·2π/a, (b) and (c) 3D distribution of light intensity and electric (red arrows) and magnetic (blue arrows) fields at the interface z=0 for k 0=0.344·2π/a and k 0=0.425·2π/a respectively.

Fig. 4.
Fig. 4.

The average energy density on xy plane as a function of z for a sample guided (k 0=0.088·2π/a, k=[0.25π/a, 0]), leaky (k 0=0.396·2π/a, k=[0.25π/a, 0]) and spurious mode (k 0=0.276 ·2π/a, k=[0.55π/a, 0]). The graph is created for a half of the structure and dotted lines mark the boundaries of the PML and of the core. The inset shows the difference between guided and the leaky mode for large z.

Fig. 5.
Fig. 5.

Eigenfrequencies and computation time for the first (a) and second (b) eigenmode of the sample structure in fig. 1 as a function of the number of planewaves.

Fig. 6.
Fig. 6.

(a) Computed photonic band graph for the structure described in the text and (b) comparison of our results (blue dotted curves) with the band diagram obtained by full 3D plane-wave expansion method (red square curves) which we reproduce from ref. [24] (fig. 8a).

Tables (1)

Tables Icon

Table 1. The computed eigenfrequencies, computation time and relative error for two first eigenmodes in sample structure as a function of the number of planewaves. The error is computed as (kN0k4410)/k4410, where kN0 is the eigenfrequency obtained for N planewaves.

Equations (48)

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z E y + y E z = i ω μ x μ 0 H x ,
x E z z E x = i ω μ y μ 0 H y ,
y E x + x E y = i ω μ z μ 0 H z ,
z H y + y H z = i ω ε x ε 0 E x ,
x H z z H x = i ω ε y ε 0 E y ,
y H x + x H y = i ω ε z ε 0 E z ,
H z = [ i k 0 η 0 μ z x i k 0 η 0 ε z y ] [ E y E x ] ,
E z = [ i η 0 k 0 ε z y i η 0 k 0 ε z x ] [ H x H y ] .
z [ E y E x ] = i η 0 k 0 [ y ε z 1 y + μ x k 0 2 y ε z 1 x x ε z 1 y x ε z 1 x + μ y k 0 2 ] [ H x H y ] ,
z [ H x H y ] = i 1 η 0 k 0 [ x μ z 1 x + ε y k 0 2 x μ z 1 y y μ z 1 x y μ z 1 y + ε x k 0 2 ] [ E y E x ] .
z E ¯ = i R ¯ H H ¯ ,
z H ¯ = i R ¯ E E ¯ .
Ψ = G Ψ G φ G ,
φ G = exp ( i G · r ) ,
φ G m φ G n = δ mn ,
Ψ ( r ) = Φ ( r ) exp ( i k · r ) ,
Ψ ( r ) = G Ψ G exp [ i ( G + k ) · r ] = G Ψ G φ G + k .
z [ E y G E x G ] = i G η 0 k 0 φ G + k φ G + k | y ε z 1 y + μ x k 0 2 y ε z 1 x x ε z 1 y x ε z 1 x + μ y k 0 2 φ G + k φ G + k [ H x G H y G ] ,
z [ H x G H y G ] = i G 1 η 0 k 0 φ G + k φ G + k x μ z 1 x + ε y k 0 2 x μ z 1 y y μ z 1 x y μ z 1 y + ε x k 0 2 φ G + k φ G + k [ E y G E x G ] .
ε x y ( r ) = G ε x y G φ G ,
ε z 1 ( r ) = G κ G φ G ,
μ x y ( r ) = G μ x y G φ G ,
μ z 1 ( r ) = G γ G φ G
z [ E y G E x G ] =
i G η 0 k 0 [ ( G y + k y ) ( G y + k y ) κ G G + k 0 2 μ x G G ' ( G y + k y ) ( G y + k x ) κ G G ( G x + k x ) ( G y + k y ) κ G G ( G x + k x ) ( G x + k x ) κ G G + k 0 2 μ y G G ] [ H x G H y G ] ,
z [ H x G H y G ] =
i G 1 η 0 k 0 [ ( G x + k x ) ( G x + k x ) γ G G + k 0 2 ε y G G ' ( G x + k x ) ( G y + k y ) γ G G ( G y + k y ) ( G x + k x ) γ G G ( G y + k y ) ( G y + k y ) γ G G + k 0 2 ε x G G ] [ E y G E x G ] ,
z 2 E ¯ = Q ¯ E E ¯ ,
z 2 H ¯ = Q ¯ H H ¯ ,
E ¯ = T E E ˜ ¯ ,
H ¯ = T H H ˜ ¯
T E 1 Q ¯ E T E = T H 1 Q ¯ H T H = Γ ¯ 2 ,
z 2 E ˜ ¯ = Γ ¯ 2 E ˜ ¯ ,
z 2 H ˜ ¯ = Γ ¯ 2 H ˜ ¯ ,
E ˜ ¯ = A cos ( Γ ¯ z ) + B sin ( Γ ¯ z ) ,
H ˜ ¯ = C cos ( Γ ¯ z ) + D sin ( Γ ¯ z ) .
H ¯ n = Y ¯ n ( k 0 , k ) E ¯ n ,
Y ¯ n = T H n { y ¯ 2 n [ y ¯ 1 n ( T H n ) 1 Y ¯ n 1 ( T E n ) ] 1 y ¯ 2 n y ¯ 1 n } ( T E n ) 1 ,
Y ¯ 1 = y ¯ 1 1 .
[ Y ¯ top ( k 0 , k ) Y ¯ bottom ( k 0 , k ) ] · E ¯ = 0 ,
ε ( r ) = ε ( r ) * exp ( r 2 σ ) ,
μ ( r ) = μ ( r ) * exp ( r 2 σ ) .
F ( k 0 ) = min eigenvalues ( [ Y ¯ top ( k 0 , k ) Y ¯ bottom ( k 0 , k ) ] ) .
P ( z ) = 1 4 [ ε 0 E ε E + μ 0 H μ H ]
= 1 4 G G [ ( E x G ) * E x G ε x G G + ( E y G ) * E y G ε y G G + ( D z G ) * D z G κ G G ]
+ 1 4 G G [ ( H x G ) * H x G μ x G G + ( H y G ) * H y G μ y G G + ( B z G ) * B z G γ G G ]
D z G = η 0 k 0 [ ( G y + k y ) H x G + ( G x + k x ) H y G ] ,
B z G = 1 η 0 k 0 [ ( G x + k x ) E y G + ( G y + k y ) E x G ] .

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