Abstract

We introduce a new implementation of the finite-difference time-domain (FDTD) algorithm with recursive convolution (RC) for first-order Drude metals. We implemented RC for both Maxwell’s equations for light polarized in the plane of incidence (TM mode) and the wave equation for light polarized normal to the plane of incidence (TE mode). We computed the Drude parameters at each wavelength using the measured value of the dielectric constant as a function of the spatial and temporal discretization to ensure both the accuracy of the material model and algorithm stability. For the TE mode, where Maxwell’s equations reduce to the wave equation (even in a region of nonuniform permittivity) we introduced a wave equation formulation of RC-FDTD. This greatly reduces the computational cost. We used our methods to compute the diffraction characteristics of metallic gratings in the visible wavelength band and compared our results with frequency-domain calculations.

© 2008 Optical Society of America

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References

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  1. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993), Chap. 8, 123-162.
  2. J. B. Cole, S. Banerjee, and M. I. Haftel, “High accuracy nonstandard finite-difference time-domain algorithms for computational electromagnetics: Applications to optics and photonics,” in Advances in the Applications of Nonstandard Finite Difference Schemes (World Scientific, 2006), pp. 89-189.
  3. A. Taflove and S. C. Hagness, Computational Electrodynamics the Finite Difference Time-Domain Method, 3rd ed. (Artech House, 2005), Chap. 8, p. 329, Chap. 9, p. 355.
  4. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Chelsea, 1987), p. 52.
  5. S. Banerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274-280 (2005).
    [CrossRef]
  6. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface relief gratings,” J. Opt. Soc. Am. A 3, 1780-1787 (1986).
    [CrossRef]
  7. J. J. Kuta, H. M. van Driel, D. Landheer, and Y. Feng, “Coupled-wave analysis of lamellar metal transmission gratings for the visible and the infrared,” J. Opt. Soc. Am. A 12, 1118-1127 (1995).
    [CrossRef]
  8. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. XIV.
  9. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 16, 4370-4379 (1972).
    [CrossRef]

2006 (1)

J. B. Cole, S. Banerjee, and M. I. Haftel, “High accuracy nonstandard finite-difference time-domain algorithms for computational electromagnetics: Applications to optics and photonics,” in Advances in the Applications of Nonstandard Finite Difference Schemes (World Scientific, 2006), pp. 89-189.

2005 (2)

A. Taflove and S. C. Hagness, Computational Electrodynamics the Finite Difference Time-Domain Method, 3rd ed. (Artech House, 2005), Chap. 8, p. 329, Chap. 9, p. 355.

S. Banerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274-280 (2005).
[CrossRef]

1999 (1)

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. XIV.

1995 (1)

1993 (1)

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993), Chap. 8, 123-162.

1987 (1)

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Chelsea, 1987), p. 52.

1986 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 16, 4370-4379 (1972).
[CrossRef]

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Chelsea, 1987), p. 52.

Banerjee, S.

J. B. Cole, S. Banerjee, and M. I. Haftel, “High accuracy nonstandard finite-difference time-domain algorithms for computational electromagnetics: Applications to optics and photonics,” in Advances in the Applications of Nonstandard Finite Difference Schemes (World Scientific, 2006), pp. 89-189.

S. Banerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274-280 (2005).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. XIV.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 16, 4370-4379 (1972).
[CrossRef]

Cole, J. B.

J. B. Cole, S. Banerjee, and M. I. Haftel, “High accuracy nonstandard finite-difference time-domain algorithms for computational electromagnetics: Applications to optics and photonics,” in Advances in the Applications of Nonstandard Finite Difference Schemes (World Scientific, 2006), pp. 89-189.

S. Banerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274-280 (2005).
[CrossRef]

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Chelsea, 1987), p. 52.

Feng, Y.

Gaylord, T. K.

Haftel, M. I.

J. B. Cole, S. Banerjee, and M. I. Haftel, “High accuracy nonstandard finite-difference time-domain algorithms for computational electromagnetics: Applications to optics and photonics,” in Advances in the Applications of Nonstandard Finite Difference Schemes (World Scientific, 2006), pp. 89-189.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics the Finite Difference Time-Domain Method, 3rd ed. (Artech House, 2005), Chap. 8, p. 329, Chap. 9, p. 355.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 16, 4370-4379 (1972).
[CrossRef]

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993), Chap. 8, 123-162.

Kuta, J. J.

Landheer, D.

Luebbers, R. J.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993), Chap. 8, 123-162.

Moharam, M. G.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics the Finite Difference Time-Domain Method, 3rd ed. (Artech House, 2005), Chap. 8, p. 329, Chap. 9, p. 355.

van Driel, H. M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. XIV.

Yatagai, T.

S. Banerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274-280 (2005).
[CrossRef]

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

Opt. Rev. (1)

S. Banerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274-280 (2005).
[CrossRef]

Phys. Rev. B (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 16, 4370-4379 (1972).
[CrossRef]

Other (5)

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999), Chap. XIV.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993), Chap. 8, 123-162.

J. B. Cole, S. Banerjee, and M. I. Haftel, “High accuracy nonstandard finite-difference time-domain algorithms for computational electromagnetics: Applications to optics and photonics,” in Advances in the Applications of Nonstandard Finite Difference Schemes (World Scientific, 2006), pp. 89-189.

A. Taflove and S. C. Hagness, Computational Electrodynamics the Finite Difference Time-Domain Method, 3rd ed. (Artech House, 2005), Chap. 8, p. 329, Chap. 9, p. 355.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Chelsea, 1987), p. 52.

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Figures (4)

Fig. 1
Fig. 1

Numerical fitting of experimental permittivity data using the first-order Drude model. The arrow indicates the point corresponding to wavelength = 704.5 nm , where the fitting is obtained.

Fig. 2
Fig. 2

Wavelength spectra of reflectivity/transmissivity for two grating configurations (1: Λ = 200 nm , t = 200 nm , 2: Λ = 400 nm , t = 400 nm ) in the TE mode, computed using RC-FDTD and RCWA. For all wavelengths, light is incident normally on the grating.

Fig. 3
Fig. 3

Wavelength spectra of reflectivity/transmissivity for grating configuration 1 ( Λ = 200 nm , t = 200 nm ) in the TM mode, computed using RC-FDTD and RCWA. (a) RC-FDTD results obtained using c 1 values given in Table 3, (b) RC-FDTD results obtained using average c 1 values (see Section 3) for wavelengths in the range 400 nm 500 nm . For all wavelengths, light is incident normally on the grating.

Fig. 4
Fig. 4

Angular spectra of reflectivity/transmissivity for grating configuration 3 ( Λ = 176.125 nm , t = 100.039 nm , λ = 704.5 nm ) computed using FDTD and RCWA (a) in the TE mode (b) in the TM mode.

Tables (3)

Tables Icon

Table 1 Refractive Indices ( η ) for Various Incident Wavelengths

Tables Icon

Table 2 Eigenvalues for TE Mode Stability

Tables Icon

Table 3 Eigenvalues for y Component Stability in the TM Mode

Equations (59)

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μ ( r ) t H ( r , t ) = × E ( r , t ) ,
t D ( r , t ) = × H ( r , t ) ,
D ( r , ω ) = ε ( r , ω ) E ( r , ω ) ,
D ( t ) = 0 t ε ( τ ) E ( t τ ) d τ ,
ε ( ω ) = 1 + ω p 2 ω ( i ν c ω ) = ε + χ ( ω ) ,
χ ( t ) = ω p 2 ν c [ 1 e ν c t ] U ( t ) ,
U ( t ) = 0 if t = 0
= 1 t > 0 .
D n = ε 0 ε E n + ε 0 m = 0 n 1 E n m m Δ t ( m + 1 ) Δ t χ ( τ ) d τ .
D n = ε 0 ε E n + ε 0 Ψ n ,
Ψ n = m = 0 n 1 E n m m Δ t ( m + 1 ) Δ t χ ( τ ) d τ = m = 0 n 1 E n m ω p 2 ν c [ Δ t + e ν c ( m + 1 ) Δ t e ν c ( m ) Δ t ν c ] .
H n + 1 2 = H n 1 2 Δ t h μ d × E n
d t D = D n + 1 D n .
E n + 1 = E n 1 ε ( Ψ n + 1 Ψ n ) + Δ t ε 0 ( ε ) h d × H n + 1 2 ,
μ ( t t D ) = 2 E ( E ) .
μ ( t t D ) = 2 E .
d t t D = D n + 1 + D n 1 2 D n .
E n + 1 = 2 E n E n 1 1 ε ( Ψ n + 1 + Ψ n 1 2 Ψ n ) + ( Δ t ) 2 μ ε h 2 d 2 E n ,
Ψ n = m = 0 n 1 E n m χ m ,
χ m = ω p 2 ν c 2 [ ν c + e m ν c ( e ν c 1 ) ] .
χ m + 1 = c 1 + c 2 χ m ,
c 1 = ω p 2 ν c ( 1 e ν c ) , c 2 = e ν c ,
Ψ 0 = 0 , Ψ 1 = E 1 χ 0 , Ψ 2 = E 2 χ 0 + c 1 E 1 + c 2 Ψ 1 .
Ψ n + 1 Ψ n = χ 0 [ E n + 1 E n ] + c 1 [ E n ] + c 2 [ Ψ n Ψ n 1 ] ,
Ψ n + 1 + Ψ n 1 2 Ψ n = χ 0 [ E n + 1 + E n 1 2 E n ] + c 1 [ E n E n 1 ] + c 2 [ Ψ n + Ψ n 2 2 Ψ n 1 ] .
ε r ( ω ) = 1 ω p 2 ν c 2 + ω 2 ,
ε i ( ω ) = ω p 2 ν c ω ( ν c 2 + ω 2 ) .
ν c = ε i ( ω 1 ) 1 ε r ( ω 1 ) ω 1 ,
ω p = ω 1 ( ν c 2 + ω 1 2 ) ε i ( ω 1 ) ν c .
ω F = 2 π ν F λ F .
ω F = λ ν F λ F ν ω .
λ = 704.5 nm , ω = 2.676 × 10 15 sec 1 , λ F = 35.227 ,
ν F = 0.538 , c 1 = 0.225 , c 2 = 1.002 .
E n + 1 = E n 1 ε ( d t Ψ n + 1 2 ) + Δ t ε 0 ( ε ) h d × H n + 1 2 ,
d t Ψ n + 3 2 = c 1 [ E n + 1 ] + c 2 [ d t Ψ n + 1 2 ] .
F n + 1 = M F n + N F n + 1 ,
F n + 1 = [ H n + 1 2 E n + 1 d t Ψ n + 3 2 ] , F n = [ H n 1 2 E n d t Ψ n + 1 2 ] .
F n + 1 = ( I N ) 1 M F n ,
M = [ 1 p 0 0 1 1 0 0 c 2 ] ,
N = [ 0 0 0 q 0 0 0 c 1 0 ] .
C = ( I N ) 1 M .
F n = C n F 0 .
F 0 = a 1 Λ 1 + a 2 Λ 2 + a 3 Λ 3 ,
F n = a 1 λ 1 n Λ 1 + a 2 λ 2 n Λ 2 + a 3 λ 3 n Λ 3 ,
( p E ) z = 1 μ Δ t h ( d x E y d y E x ) .
( q H ) x = 1 ε 0 ε Δ t h ( d × H ) x = 1 ε 0 ε Δ t h ( d y H z ) ,
( q H ) y = 1 ε 0 ε Δ t h ( d × H ) y = 1 ε 0 ε Δ t h ( d x H z ) .
d l G = ( e i k l h 2 e i k l h 2 ) G .
k x = 2 π η λ F cos ( θ ) ,
k y = 2 π η λ F sin ( θ ) ,
d t t Ψ n = c 1 [ d t E n 1 2 ] + c 2 [ d t t Ψ n 1 ] ,
E n + 1 = d t E n 1 2 1 ε ( d t t Ψ n ) + ( 1 + b ) E n .
d t E n + 1 2 = E n + 1 E n .
F n + 1 = [ d t t Ψ n E n + 1 d t E n + 1 2 ] ,
F n = [ d t t Ψ n 1 E n d t E n 1 2 ] ,
M = [ c 2 0 c 1 0 1 + b 1 0 1 0 ] ,
N = [ 0 0 0 1 0 0 0 1 0 ] .
d 2 E z = [ ( e i k x h + e i k x h 2 ) + ( e i k y h + e i k y h 2 ) ] E z .
b E z = ( Δ t ) 2 μ ε ( h ) 2 [ ( e i k x h + e i k x h 2 ) + ( e i k y h + e i k y h 2 ) ] E z .

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