Abstract

In this Letter, we present an efficient complex-envelope alternating-direction-implicit finite-difference time-domain (CE-ADI-FDTD) method for the transient analysis of magnetic photonic crystals with lossy ferrites. The proposed CE-ADI-FDTD method is generally formulated for a saturated ferrite with anisotropic permittivity tensor and ferrite loss. Auxiliary differential equations for modeling saturated ferrite and Maxwell’s curl equations are first cast into a first-order differential system in a CE form. Then, by using an efficient ADI splitting formulas, the proposed CE-ADI-FDTD method is attained in a very concise form with few and simple right-hand side terms. The performance of the proposed method is validated and compared with the explicit FDTD method.

© 2011 Optical Society of America

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  1. A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
    [CrossRef]
  2. A. Figotin and I. Vitebskiy, Phys. Rev. A 76, 053839 (2007).
    [CrossRef]
  3. K.-Y. Jung and F. L. Teixeira, Phys. Rev. A 78, 043826(2008).
    [CrossRef]
  4. E. L. Tan, IEEE Antennas Propag Mag. 56, 170 (2008).
    [CrossRef]
  5. H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., IEEE Photon. Technol. Lett. 14, 477 (2002).
    [CrossRef]
  6. S. Chen, W. Zang, A. Schulzgen, J. Liu, L. Han, Y. Zeng, J. Tian, F. Song, J. V. Moloney, and N. Peyghambarian, Opt. Lett. 33, 2755 (2008).
    [CrossRef] [PubMed]
  7. R. Zhu, S. Fu, and H. Peng, J. Magn. Magn. Mater. 323, 145 (2011).
  8. A. B. Khanikaev and M. J. Steel, Opt. Express 17, 5265(2009).
    [CrossRef] [PubMed]
  9. K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer, 2007).
  10. K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
    [CrossRef]

2011 (1)

R. Zhu, S. Fu, and H. Peng, J. Magn. Magn. Mater. 323, 145 (2011).

2009 (1)

2008 (3)

2007 (1)

A. Figotin and I. Vitebskiy, Phys. Rev. A 76, 053839 (2007).
[CrossRef]

2006 (1)

K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
[CrossRef]

2003 (1)

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

2002 (1)

H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., IEEE Photon. Technol. Lett. 14, 477 (2002).
[CrossRef]

Chen, S.

Donderici, B.

K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, Phys. Rev. A 76, 053839 (2007).
[CrossRef]

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

Fu, S.

R. Zhu, S. Fu, and H. Peng, J. Magn. Magn. Mater. 323, 145 (2011).

Han, L.

Jung, K.-Y.

K.-Y. Jung and F. L. Teixeira, Phys. Rev. A 78, 043826(2008).
[CrossRef]

K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
[CrossRef]

Khanikaev, A. B.

Li, D.

K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer, 2007).

Liu, J.

Moloney, J. V.

Osgood, R. M.

H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., IEEE Photon. Technol. Lett. 14, 477 (2002).
[CrossRef]

Peng, H.

R. Zhu, S. Fu, and H. Peng, J. Magn. Magn. Mater. 323, 145 (2011).

Peyghambarian, N.

Rao, H.

H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., IEEE Photon. Technol. Lett. 14, 477 (2002).
[CrossRef]

Scarmozzino, R.

H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., IEEE Photon. Technol. Lett. 14, 477 (2002).
[CrossRef]

Schulzgen, A.

Song, F.

Steel, M. J.

Tan, E. L.

E. L. Tan, IEEE Antennas Propag Mag. 56, 170 (2008).
[CrossRef]

Teixeira, F. L.

K.-Y. Jung and F. L. Teixeira, Phys. Rev. A 78, 043826(2008).
[CrossRef]

K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
[CrossRef]

Tian, J.

Vitebskiy, I.

A. Figotin and I. Vitebskiy, Phys. Rev. A 76, 053839 (2007).
[CrossRef]

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

Zang, W.

Zeng, Y.

Zhang, K.

K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer, 2007).

Zhu, R.

R. Zhu, S. Fu, and H. Peng, J. Magn. Magn. Mater. 323, 145 (2011).

IEEE Antennas Propag Mag. (1)

E. L. Tan, IEEE Antennas Propag Mag. 56, 170 (2008).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

H. Rao, R. Scarmozzino, and R. M. Osgood, Jr., IEEE Photon. Technol. Lett. 14, 477 (2002).
[CrossRef]

J. Magn. Magn. Mater. (1)

R. Zhu, S. Fu, and H. Peng, J. Magn. Magn. Mater. 323, 145 (2011).

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (2)

A. Figotin and I. Vitebskiy, Phys. Rev. A 76, 053839 (2007).
[CrossRef]

K.-Y. Jung and F. L. Teixeira, Phys. Rev. A 78, 043826(2008).
[CrossRef]

Phys. Rev. B (2)

A. Figotin and I. Vitebskiy, Phys. Rev. B 67, 165210 (2003).
[CrossRef]

K.-Y. Jung, B. Donderici, and F. L. Teixeira, Phys. Rev. B 74, 165207 (2006).
[CrossRef]

Other (1)

K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer, 2007).

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Figures (2)

Fig. 1
Fig. 1

(a) Band diagram and (b) transmittance plots for lossless magnetic PhC. Transmittance determined with incident electric field polarized at ϕ = 32.5 ° .

Fig. 2
Fig. 2

Normalized | E | 2 versus position for (a) lossless and (b) lossy ferrite cases. Markers ○, +, and □ attained with proposed CE-ADI-FDTD method at CFLN of 100, 150, and 200, respectively. Continuous line plot attained with explicit FDTD method at CFLN of 1.

Equations (25)

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E t = κ × H ,
H t = 1 μ 0 × E M t ,
M t = γ μ 0 ( M s z ^ × H + M × H 0 z ^ ) + α M s ( M s z ^ × M t ) ,
U ˜ t = ( j ω c I + M ) U ˜ ,
U ˜ = [ E x ˜ E y ˜ H x ˜ H y ˜ M x ˜ M y ˜ ] T ,
Sub-step   1 : V ^ n = G U ˜ n V ^ n 1 2 ,
( I Δ t 4 A ) U ˜ n + 1 2 = V ^ n ,
Sub-step   2 : V ^ n + 1 2 = G U ˜ n + 1 2 V ^ n ,
( I Δ t 4 B ) U ˜ n + 1 = V ^ n + 1 2 ,
V ^ = [ e x ^ e y ^ h x ^ h y ^ m x ^ m y ^ ] T ,
A = [ 0 0 κ x y z 1 2 + j ω c Δ t 8 0 0 0 0 0 κ y y z 1 2 + j ω c Δ t 8 0 0 0 0 p 11 μ 0 z 0 0 0 0 0 p 21 μ 0 z 0 0 0 0 0 p 31 μ 0 z 0 0 0 0 0 p 41 μ 0 z 0 0 0 0 ] ,
B = [ 0 0 0 κ x x z 1 2 + j ω c Δ t 8 0 0 0 0 0 κ y x z 1 2 + j ω c Δ t 8 0 0 p 12 μ 0 z 0 0 0 0 0 p 22 μ 0 z 0 0 0 0 0 p 32 μ 0 z 0 0 0 0 0 p 42 μ 0 z 0 0 0 0 0 ] ,
G = [ ( 1 1 2 + j ω c Δ t 8 ) I 0 0 P ] .
P = [ 1 2 + Δ t ξ 1 4 Δ t ξ 2 4 Δ t ξ 3 4 Δ t ξ 4 4 Δ t ξ 2 4 1 2 + Δ t ξ 1 4 Δ t ξ 4 4 Δ t ξ 3 4 Δ t ξ 5 4 Δ t ξ 2 4 1 2 + Δ t ξ 6 4 Δ t ξ 4 4 Δ t ξ 2 4 Δ t ξ 5 4 Δ t ξ 4 4 1 2 + Δ t ξ 6 4 ] 1 ,
ξ 1 = j ω c 2 + ω m α 2 ( α 2 + 1 ) , ξ 2 = ω m 2 ( α 2 + 1 ) , ξ 3 = ω 0 α 2 ( α 2 + 1 ) , ξ 6 = j ω c 2 + ω 0 α 2 ( α 2 + 1 ) , ξ 4 = ω 0 2 ( α 2 + 1 ) , ξ 5 = ω m α 2 ( α 2 + 1 ) ,
e x ^ k n = 1 1 2 + j ω c Δ t 8 E x ˜ k n e x ^ k n 1 2 ,
e y ^ k n = 1 1 2 + j ω c Δ t 8 E y ˜ k n e y ^ k n 1 2 ,
[ h x ^ h y ^ m x ^ m y ^ ] k + 1 2 n = P [ H x ˜ H y ˜ M x ˜ M y ˜ ] k + 1 2 n [ h x ^ h y ^ m x ^ m y ^ ] k + 1 2 n 1 2 .
( p 11 κ y y Δ t 2 16 μ 0 Δ z 2 1 2 + j ω c Δ t 8 ) ( E y ˜ k 1 n + 1 2 + E y ˜ k + 1 n + 1 2 ) + ( 1 + 2 ( p 11 κ y y Δ t 2 16 μ 0 Δ z 2 1 2 + j ω c Δ t 8 ) ) × E y ˜ k n + 1 2 = e y ^ k n + ( κ y y Δ t 4 Δ z 1 2 + j ω c Δ t 8 ) ( h x ^ k + 1 2 n h x ^ k 1 2 n ) .
H x ˜ k + 1 2 n + 1 2 = h x ^ k + 1 2 n + ( p 11 Δ t 4 μ 0 Δ z ) ( E y ˜ k + 1 n + 1 2 E y ˜ k n + 1 2 ) ,
H y ˜ k + 1 2 n + 1 2 = h y ^ k + 1 2 n + ( p 21 Δ t 4 μ 0 Δ z ) ( E y ˜ k + 1 n + 1 2 E y ˜ k n + 1 2 ) ,
M x ˜ k + 1 2 n + 1 2 = m x ^ k + 1 2 n + ( p 31 Δ t 4 μ 0 Δ z ) ( E y ˜ k + 1 n + 1 2 E y ˜ k n + 1 2 ) ,
M y ˜ k + 1 2 n + 1 2 = m y ^ k + 1 2 n + ( p 41 Δ t 4 μ 0 Δ z ) ( E y ˜ k + 1 n + 1 2 E y ˜ k n + 1 2 ) ,
E x ˜ k n + 1 2 = e x ^ k n + ( κ x y Δ t 4 Δ z 1 2 + j ω c Δ t 8 ) ( H x ˜ k + 1 2 n + 1 2 H x ˜ k 1 2 n + 1 2 ) .
ϵ = ϵ 0 [ ϵ A + δ A cos ( 2 ϕ A ) δ A sin ( 2 ϕ A ) 0 δ A sin ( 2 ϕ A ) ϵ A δ A cos ( 2 ϕ A ) 0 0 0 ϵ z z ] .

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