Abstract

The nonstandard (NS) finite-difference time domain (FDTD) algorithm has proved be remarkably accurate on a coarse numerical grid, but the well-known resonances called whispering gallery modes (WGMs) in the Mie regime are very sensitive to the scatterer representation on the computational grid, and a very large number of time steps are needed to correctly calculate the modes because the electromagnetic field outside the scatterer is weakly coupled to the inside. Using the NS-FDTD algorithm on a coarse grid, we were able to accurately simulate the WGMs of dielectric cylinders in the Mie regime.

© 2010 Optical Society of America

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References

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  1. L. Rayleigh, “The problem of the whispering gallery,” Philos. Mag. 20, 1001-1004 (1910).
  2. P. Debye, “Der Lichtdruck auf Kugeln von Beliebigem Material,” Ann. Phys. 30, 57-136 (1909).
    [CrossRef]
  3. C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering gallery modes of spheres,” Phys. Rev. 124, 1807-1809 (1961).
    [CrossRef]
  4. P. Chyek, V. Ramaswamy, A. Ashkin, and J. M. Dziedzic, “Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data,” Appl. Opt. 22, 2302-2307 (1983).
    [CrossRef]
  5. A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering gallery mode resonators in photonics and nonlinear optics,” IPN Progress Report, 42-162 (2005).
  6. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1989).
  7. K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
    [CrossRef]
  8. J. B. Cole, “High-accuracy Yee algorithm based on nonstandard finite differences: new developments and verifications,” IEEE Trans. Antennas Propag. 50, 1185-1191 (2002).
    [CrossRef]
  9. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  10. S. K. Godunov, Difference Schemes: An Introduction to the Underlying Theory (North-Holland, 1987).
  11. R. E. Mickens, Nonstandard Finite Difference Models of Differential Equation (World Scientific, 1994).
  12. J. B. Cole and D. Zhu, “Improved version of the second-order Mur absorbing boundary condition based on a nonstandard finite difference model,” Appl. Comput. Electromagn. Soc. J. 24(4) (2009).
  13. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972-2974 (2006).
    [CrossRef] [PubMed]
  14. S. E. Krakiwsky, L. E. Turner, and M. M. Okoniewski, “Acceleration of finite-difference time domain (FDTD) using graphics processor units (GPU),” in IEEE MTT-S International Microwave Symposium Digest (IEEE, 2004), Vol. 2, pp. 1033-1036.

2009 (1)

J. B. Cole and D. Zhu, “Improved version of the second-order Mur absorbing boundary condition based on a nonstandard finite difference model,” Appl. Comput. Electromagn. Soc. J. 24(4) (2009).

2006 (1)

2002 (1)

J. B. Cole, “High-accuracy Yee algorithm based on nonstandard finite differences: new developments and verifications,” IEEE Trans. Antennas Propag. 50, 1185-1191 (2002).
[CrossRef]

1994 (1)

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

1983 (1)

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

1961 (1)

C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering gallery modes of spheres,” Phys. Rev. 124, 1807-1809 (1961).
[CrossRef]

1910 (1)

L. Rayleigh, “The problem of the whispering gallery,” Philos. Mag. 20, 1001-1004 (1910).

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von Beliebigem Material,” Ann. Phys. 30, 57-136 (1909).
[CrossRef]

Ashkin, A.

Barber, P. W.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1989).

Berenger, J. P.

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

Bermel, P.

Bond, W. L.

C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering gallery modes of spheres,” Phys. Rev. 124, 1807-1809 (1961).
[CrossRef]

Burr, G.

Chyek, P.

Cole, J. B.

J. B. Cole and D. Zhu, “Improved version of the second-order Mur absorbing boundary condition based on a nonstandard finite difference model,” Appl. Comput. Electromagn. Soc. J. 24(4) (2009).

J. B. Cole, “High-accuracy Yee algorithm based on nonstandard finite differences: new developments and verifications,” IEEE Trans. Antennas Propag. 50, 1185-1191 (2002).
[CrossRef]

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von Beliebigem Material,” Ann. Phys. 30, 57-136 (1909).
[CrossRef]

Dziedzic, J. M.

Farjadpour, A.

Garrett, C. G. B.

C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering gallery modes of spheres,” Phys. Rev. 124, 1807-1809 (1961).
[CrossRef]

Godunov, S. K.

S. K. Godunov, Difference Schemes: An Introduction to the Underlying Theory (North-Holland, 1987).

Hill, S. C.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1989).

Ibanescu, M.

Ilchenko, V. S.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering gallery mode resonators in photonics and nonlinear optics,” IPN Progress Report, 42-162 (2005).

Joannopoulos, J. D.

Johnson, S. G.

Kaiser, W.

C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering gallery modes of spheres,” Phys. Rev. 124, 1807-1809 (1961).
[CrossRef]

Krakiwsky, S. E.

S. E. Krakiwsky, L. E. Turner, and M. M. Okoniewski, “Acceleration of finite-difference time domain (FDTD) using graphics processor units (GPU),” in IEEE MTT-S International Microwave Symposium Digest (IEEE, 2004), Vol. 2, pp. 1033-1036.

Maleki, L.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering gallery mode resonators in photonics and nonlinear optics,” IPN Progress Report, 42-162 (2005).

Matsko, A. B.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering gallery mode resonators in photonics and nonlinear optics,” IPN Progress Report, 42-162 (2005).

Mickens, R. E.

R. E. Mickens, Nonstandard Finite Difference Models of Differential Equation (World Scientific, 1994).

Okoniewski, M. M.

S. E. Krakiwsky, L. E. Turner, and M. M. Okoniewski, “Acceleration of finite-difference time domain (FDTD) using graphics processor units (GPU),” in IEEE MTT-S International Microwave Symposium Digest (IEEE, 2004), Vol. 2, pp. 1033-1036.

Ramaswamy, V.

Rayleigh, L.

L. Rayleigh, “The problem of the whispering gallery,” Philos. Mag. 20, 1001-1004 (1910).

Rodriguez, A.

Roundy, D.

Savchenkov, A. A.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering gallery mode resonators in photonics and nonlinear optics,” IPN Progress Report, 42-162 (2005).

Strekalov, D.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering gallery mode resonators in photonics and nonlinear optics,” IPN Progress Report, 42-162 (2005).

Turner, L. E.

S. E. Krakiwsky, L. E. Turner, and M. M. Okoniewski, “Acceleration of finite-difference time domain (FDTD) using graphics processor units (GPU),” in IEEE MTT-S International Microwave Symposium Digest (IEEE, 2004), Vol. 2, pp. 1033-1036.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

Zhu, D.

J. B. Cole and D. Zhu, “Improved version of the second-order Mur absorbing boundary condition based on a nonstandard finite difference model,” Appl. Comput. Electromagn. Soc. J. 24(4) (2009).

Ann. Phys. (1)

P. Debye, “Der Lichtdruck auf Kugeln von Beliebigem Material,” Ann. Phys. 30, 57-136 (1909).
[CrossRef]

Appl. Comput. Electromagn. Soc. J. (1)

J. B. Cole and D. Zhu, “Improved version of the second-order Mur absorbing boundary condition based on a nonstandard finite difference model,” Appl. Comput. Electromagn. Soc. J. 24(4) (2009).

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (2)

K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966).
[CrossRef]

J. B. Cole, “High-accuracy Yee algorithm based on nonstandard finite differences: new developments and verifications,” IEEE Trans. Antennas Propag. 50, 1185-1191 (2002).
[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]

Opt. Lett. (1)

Philos. Mag. (1)

L. Rayleigh, “The problem of the whispering gallery,” Philos. Mag. 20, 1001-1004 (1910).

Phys. Rev. (1)

C. G. B. Garrett, W. Kaiser, and W. L. Bond, “Stimulated emission into optical whispering gallery modes of spheres,” Phys. Rev. 124, 1807-1809 (1961).
[CrossRef]

Other (5)

S. E. Krakiwsky, L. E. Turner, and M. M. Okoniewski, “Acceleration of finite-difference time domain (FDTD) using graphics processor units (GPU),” in IEEE MTT-S International Microwave Symposium Digest (IEEE, 2004), Vol. 2, pp. 1033-1036.

A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Review of applications of whispering gallery mode resonators in photonics and nonlinear optics,” IPN Progress Report, 42-162 (2005).

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1989).

S. K. Godunov, Difference Schemes: An Introduction to the Underlying Theory (North-Holland, 1987).

R. E. Mickens, Nonstandard Finite Difference Models of Differential Equation (World Scientific, 1994).

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Figures (10)

Fig. 1
Fig. 1

Infinite plane wave impinging on an infinite dielectric cylinder ( a = radius , k = wave vector). TM and TE polarizations are shown. Wave propagates along the + x axis.

Fig. 2
Fig. 2

Expansion coefficients for a resonance. Absolute values of (a) d l and (b) b l , respectively.

Fig. 3
Fig. 3

(a) Staircase model. A and C are inside (gray), B and D are outside (white) the scatterer. (b)–(d) Circle of radius 6 h ( h = grid size) centered at [ ( x + s ) h , ( y + s ) h ] ( x , y are integers) for shift parameters s=0.0, 0.25, 0.5 for (b), (c), (d), respectively.

Fig. 4
Fig. 4

(a) Integration of H on contour about E z grid point. (b)–(d) Effect of shifting the center of a circle of radius 6 h . Center at [ ( x + s ) h , ( y + s ) h ] ( x , y are integers). x , y ε ( x , y ) is invariant with respect to shifts s=0.0, 0.25, 0.5 for (b), (c), (d), respectively.

Fig. 5
Fig. 5

Fuzzy model for the TE mode. Closed curves centered at (a) E x and (b) E y for Ampére’s law.

Fig. 6
Fig. 6

FDTD calculation of | E z s | 2 in the TM mode. (a) Analytic solution. (b), (c) Simulation results by S-FDTD and NS-FDTD algorithms at 100,000 time steps, respectively. (d) Angular intensity distributions.

Fig. 7
Fig. 7

FDTD calculation of | E y s | 2 in the TE mode. (a) Analytic solution. (b), (c) Simulation results by S-FDTD and NS-FDTD algorithms at 100,000 time steps, respectively. (d) Angular intensity distributions.

Fig. 8
Fig. 8

Convergence time for resonance and off-resonance modes. The ordinate is the rms error relative to Mie theory.

Fig. 9
Fig. 9

Whispering gallery mode excitation using a point source. (a)–(c) E z s on a color scale ranging from blue (minus) to red (plus), at 200, 1000, and 2000 time steps. The white dot is a point source. (d) Rise in surface intensity with the infinite plane wave and point source excitation.

Fig. 10
Fig. 10

RMS error in the angular distribution of scattered intensity as a function of grid fineness (a) at 100,000 × ( λ h ) 64 time steps in the TM mode, (b) in the TE mode.

Tables (3)

Tables Icon

Table 1 Examples of Resonance Conditions for TM and TE Modes a

Tables Icon

Table 2 Maximum Values of ω Δ t k h for S-FDTD and NS-FDTD Algorithm Stability

Tables Icon

Table 3 Computational Parameters of Whispering Gallery Mode Simulation in the Mie Regime

Equations (52)

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( 2 + k 2 ) E z = 0 ,
E z s ( r , θ ) = l = i l b l H l ( 1 ) ( k r ) e i l θ ,
E z i ( r , θ ) = l = i l d l J l ( n k r ) e i l θ ,
E z 0 = e i k x = l = i l J l ( k r ) e i l θ ,
E z i ( r , θ ) = E z 0 ( a , θ ) + E z s ( a , θ ) ,
r E z i ( r , θ ) = r E z 0 ( a , θ ) + r E z s ( a , θ ) .
b l ( x , n ) = n J l ( n x ) J l ( x ) J l ( n x ) J l ( x ) n J l ( n x ) H l ( 1 ) ( x ) J l ( n x ) H l ( 1 ) ( x ) ,
d l ( x , n ) = J l ( x ) b l ( x ) H l ( 1 ) ( x ) J l ( n x ) = J l 1 ( x ) H l ( 1 ) ( x ) J l ( x ) H l 1 ( 1 ) ( x ) b l d ( x ) ,
J l 1 ( n k a ) J l ( n k a ) = 1 n H l 1 ( 1 ) ( k a ) H l ( 1 ) ( k a ) .
J l 2 ( n k a ) J l 1 ( n k a ) = n H l 2 ( 1 ) ( k a ) H l 1 ( 1 ) ( k a ) ( l 1 ) ( n 2 1 ) n k a .
μ t H = × E ,
ε t E = × H J ,
d f ( x ) d x d x f ( x ) Δ x ,
d = d x x ̂ + d y y ̂ + d z z ̂ ,
d t H t = Δ t μ h d × E t ,
d t E t + Δ t 2 = Δ t ε h d × H t + Δ t 2 Δ t ε J t + Δ t 2 ,
H t + Δ t 2 = H t Δ t 2 Δ t μ h d × E t ,
E t + Δ t = E t + Δ t ε h d × H t + Δ t 2 Δ t ε J t + Δ t 2 .
μ 0 t H = × E ,
ε 0 t E = × H ,
μ 0 t H s = × E s ,
ε t E s = × H s ( ε ε 0 ) t E 0 .
J ( r , t ) = ( ε ε 0 ) Θ ( t ) I [ i ω E 0 ( r , t ) ] ,
ω Δ t k h 1 D ,
h h 2 ϵ S ϵ S 4 .
h h 2 C S 2 D + 1 C S .
d f ( x ) d x d x f ( x ) s ( Δ x ) ,
s ( Δ x ) = d x f ( x ) f ( x )
lim Δ x 0 s ( Δ x ) = Δ x .
s ( Δ x ) = s ( 0 ) + Δ x s ( 0 ) + .
s ( k , Δ x ) = 2 k sin ( k Δ x 2 )
( t 2 c 2 2 ) ψ ( r , t ) = 0 ,
d 0 = d + 1 γ 4 ( x ̂ d x ( d y 2 + d z 2 + η 3 d y 2 d z 2 ) + y ̂ d y ( d x 2 + d z 2 + η 3 d x 2 d z 2 ) + z ̂ d z ( d x 2 + d y 2 + η 3 d x 2 d y 2 ) ) ,
γ = 2 3 1 90 ( k h ) 2 ,
η = 2 5 + ( 1913 50400 5 2 252 ) ( k h ) 2 + .
ϵ S = 1 φ ( 2 ( d d ) h 2 ) φ k 2 ( k h ) 2 12 sin 2 2 θ .
ϵ NS = 1 φ ( 2 ( d 0 d ) s ( k , h ) 2 ) φ k 2 ( k h ) 6 24192 ( 2 1 sin 2 2 θ 2 ) sin 2 2 θ .
H μ t + Δ t 2 = H μ t Δ t 2 d × E t ,
E t + Δ t = E t + u 0 2 ε ̃ d 0 × H μ t + Δ t 2 Δ t ε J NS t + Δ t 2 ,
ω Δ t k h 2 D π arcsin ( 2 M D sin ( π 2 D ) ) ,
C H d s = S ε t E d S .
C H d s = S ( × H ) d S .
( × H ) z ε x y t E z ,
ε x y = ε 1 ( S Δ x Δ y ) + ε 2 ( 1 S Δ x Δ y ) .
ε t E x = y H z .
ε ( r ) y = y y + Δ y ε ( x , y ) d y = ε 1 ( L Δ y ) + ε 2 ( 1 L Δ y ) .
D = ε E , D = 0
( ε E ) = E ε + ε E = 0 .
ϵ S ( λ h ) ϵ ̃ A S ( λ h ) + ϵ ̃ S S ( λ h ) ,
ϵ NS ( λ h ) ϵ ̃ A NS ( λ h ) + ϵ ̃ S NS ( λ h ) .
ϵ ̃ S NS ( 32 ) ϵ NS ( 32 ) 0.384 .
min [ ϵ ̃ S S ( 32 ) ] = ϵ S ( 32 ) ϵ S ( 16 ) 4 6.368 .

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