Abstract

In this paper we represent a number of new physical results obtained using time domain methods and based on equivalent replacement of initially open electrodynamic problems with closed ones. These results prove the high efficiency and reliability of the approach, being grounded in our companion paper in this issue.

© 2010 Optical Society of America

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References

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  1. K. Y. Sirenko, Y. K. Sirenko, and N. P. Yashina, “Modeling and analysis of transients in periodic gratings. I. Fully absorbing boundaries for 2-D open problems,” J. Opt. Soc. Am. A 27, 532-543 (2010).
    [CrossRef]
  2. A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).
  3. Y. K. Sirenko, S. Strom, and N. P. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques (Springer, 2007).
    [PubMed]
  4. O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Springer-Verlag, 1985).
  5. B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629-651 (1977).
    [CrossRef]
  6. G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic-field equations,” Int. J. Remote Sens. 23, 377-382 (1981).
  7. Y. K. Sirenko and N. P. Yashina, “Nonstationary model problems for waveguide open resonator theory,” Electromagnetics 19, 419-442 (1999).
    [CrossRef]
  8. Y. K. Sirenko and N. P. Yashina, “Time domain theory of open waveguide resonators: canonical problems and a generalized matrix technique,” Radio Sci. 38, VIC 26-1-VIC 26-12 (2003).
    [CrossRef]
  9. V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkov State Univ. Press, 1973) (in Russian).
  10. R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
    [CrossRef]
  11. V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, “Diffraction gratings,” in Resonance Wave Scattering, Vol. 1 (Naukova Dumka, 1986) (in Russian).
  12. V. P. Shestopalov and Y. K. Sirenko, Dynamic Theory of Gratings (Naukova Dumka, 1989) (in Russian).
  13. L. G. Velychko, Y. K. Sirenko, and O. S. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” PIER 61, 1-26 (2006).
    [CrossRef]
  14. A. O. Perov, Y. K. Sirenko, and N. P. Yashina, “Periodic open resonators: peculiarities of pulse scattering and spectral features,” PIER 46, 33-75 (2004).
    [CrossRef]
  15. V. P. Shestopalov, A. A. Kirilenko, and L. A. Rud', “Waveguide discontinuities,” in Resonance Wave Scattering, Vol. 2 (Naukova Dumka, 1986) (in Russian).
  16. K. Y. Sirenko, “Splitting of super-broadband pulses by simple inhomogeneities of circular and coaxial waveguide,” Telecommun. Radio Eng. 67, 1415-1428 (2008).
    [CrossRef]

2010 (1)

2008 (1)

K. Y. Sirenko, “Splitting of super-broadband pulses by simple inhomogeneities of circular and coaxial waveguide,” Telecommun. Radio Eng. 67, 1415-1428 (2008).
[CrossRef]

2006 (1)

L. G. Velychko, Y. K. Sirenko, and O. S. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” PIER 61, 1-26 (2006).
[CrossRef]

2004 (1)

A. O. Perov, Y. K. Sirenko, and N. P. Yashina, “Periodic open resonators: peculiarities of pulse scattering and spectral features,” PIER 46, 33-75 (2004).
[CrossRef]

2003 (1)

Y. K. Sirenko and N. P. Yashina, “Time domain theory of open waveguide resonators: canonical problems and a generalized matrix technique,” Radio Sci. 38, VIC 26-1-VIC 26-12 (2003).
[CrossRef]

1999 (1)

Y. K. Sirenko and N. P. Yashina, “Nonstationary model problems for waveguide open resonator theory,” Electromagnetics 19, 419-442 (1999).
[CrossRef]

1981 (1)

G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic-field equations,” Int. J. Remote Sens. 23, 377-382 (1981).

1977 (1)

B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629-651 (1977).
[CrossRef]

Engquist, B.

B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629-651 (1977).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).

Kirilenko, A. A.

V. P. Shestopalov, A. A. Kirilenko, and L. A. Rud', “Waveguide discontinuities,” in Resonance Wave Scattering, Vol. 2 (Naukova Dumka, 1986) (in Russian).

V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, “Diffraction gratings,” in Resonance Wave Scattering, Vol. 1 (Naukova Dumka, 1986) (in Russian).

Ladyzhenskaya, O. A.

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Springer-Verlag, 1985).

Litvinenko, L. N.

V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkov State Univ. Press, 1973) (in Russian).

Majda, A.

B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629-651 (1977).
[CrossRef]

Masalov, S. A.

V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkov State Univ. Press, 1973) (in Russian).

V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, “Diffraction gratings,” in Resonance Wave Scattering, Vol. 1 (Naukova Dumka, 1986) (in Russian).

Mur, G.

G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic-field equations,” Int. J. Remote Sens. 23, 377-382 (1981).

Perov, A. O.

A. O. Perov, Y. K. Sirenko, and N. P. Yashina, “Periodic open resonators: peculiarities of pulse scattering and spectral features,” PIER 46, 33-75 (2004).
[CrossRef]

Rud', L. A.

V. P. Shestopalov, A. A. Kirilenko, and L. A. Rud', “Waveguide discontinuities,” in Resonance Wave Scattering, Vol. 2 (Naukova Dumka, 1986) (in Russian).

Shafalyuk, O. S.

L. G. Velychko, Y. K. Sirenko, and O. S. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” PIER 61, 1-26 (2006).
[CrossRef]

Shestopalov, V. P.

V. P. Shestopalov, A. A. Kirilenko, and L. A. Rud', “Waveguide discontinuities,” in Resonance Wave Scattering, Vol. 2 (Naukova Dumka, 1986) (in Russian).

V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, “Diffraction gratings,” in Resonance Wave Scattering, Vol. 1 (Naukova Dumka, 1986) (in Russian).

V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkov State Univ. Press, 1973) (in Russian).

V. P. Shestopalov and Y. K. Sirenko, Dynamic Theory of Gratings (Naukova Dumka, 1989) (in Russian).

Sirenko, K. Y.

K. Y. Sirenko, Y. K. Sirenko, and N. P. Yashina, “Modeling and analysis of transients in periodic gratings. I. Fully absorbing boundaries for 2-D open problems,” J. Opt. Soc. Am. A 27, 532-543 (2010).
[CrossRef]

K. Y. Sirenko, “Splitting of super-broadband pulses by simple inhomogeneities of circular and coaxial waveguide,” Telecommun. Radio Eng. 67, 1415-1428 (2008).
[CrossRef]

Sirenko, Y. K.

K. Y. Sirenko, Y. K. Sirenko, and N. P. Yashina, “Modeling and analysis of transients in periodic gratings. I. Fully absorbing boundaries for 2-D open problems,” J. Opt. Soc. Am. A 27, 532-543 (2010).
[CrossRef]

L. G. Velychko, Y. K. Sirenko, and O. S. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” PIER 61, 1-26 (2006).
[CrossRef]

A. O. Perov, Y. K. Sirenko, and N. P. Yashina, “Periodic open resonators: peculiarities of pulse scattering and spectral features,” PIER 46, 33-75 (2004).
[CrossRef]

Y. K. Sirenko and N. P. Yashina, “Time domain theory of open waveguide resonators: canonical problems and a generalized matrix technique,” Radio Sci. 38, VIC 26-1-VIC 26-12 (2003).
[CrossRef]

Y. K. Sirenko and N. P. Yashina, “Nonstationary model problems for waveguide open resonator theory,” Electromagnetics 19, 419-442 (1999).
[CrossRef]

V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, “Diffraction gratings,” in Resonance Wave Scattering, Vol. 1 (Naukova Dumka, 1986) (in Russian).

Y. K. Sirenko, S. Strom, and N. P. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques (Springer, 2007).
[PubMed]

V. P. Shestopalov and Y. K. Sirenko, Dynamic Theory of Gratings (Naukova Dumka, 1989) (in Russian).

Sologub, V. G.

V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkov State Univ. Press, 1973) (in Russian).

Strom, S.

Y. K. Sirenko, S. Strom, and N. P. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques (Springer, 2007).
[PubMed]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).

Velychko, L. G.

L. G. Velychko, Y. K. Sirenko, and O. S. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” PIER 61, 1-26 (2006).
[CrossRef]

Yashina, N. P.

K. Y. Sirenko, Y. K. Sirenko, and N. P. Yashina, “Modeling and analysis of transients in periodic gratings. I. Fully absorbing boundaries for 2-D open problems,” J. Opt. Soc. Am. A 27, 532-543 (2010).
[CrossRef]

A. O. Perov, Y. K. Sirenko, and N. P. Yashina, “Periodic open resonators: peculiarities of pulse scattering and spectral features,” PIER 46, 33-75 (2004).
[CrossRef]

Y. K. Sirenko and N. P. Yashina, “Time domain theory of open waveguide resonators: canonical problems and a generalized matrix technique,” Radio Sci. 38, VIC 26-1-VIC 26-12 (2003).
[CrossRef]

Y. K. Sirenko and N. P. Yashina, “Nonstationary model problems for waveguide open resonator theory,” Electromagnetics 19, 419-442 (1999).
[CrossRef]

Y. K. Sirenko, S. Strom, and N. P. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques (Springer, 2007).
[PubMed]

Electromagnetics (1)

Y. K. Sirenko and N. P. Yashina, “Nonstationary model problems for waveguide open resonator theory,” Electromagnetics 19, 419-442 (1999).
[CrossRef]

Int. J. Remote Sens. (1)

G. Mur, “Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic-field equations,” Int. J. Remote Sens. 23, 377-382 (1981).

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

Math. Comput. (1)

B. Engquist and A. Majda, “Absorbing boundary conditions for the numerical simulation of waves,” Math. Comput. 31, 629-651 (1977).
[CrossRef]

PIER (2)

L. G. Velychko, Y. K. Sirenko, and O. S. Shafalyuk, “Time-domain analysis of open resonators. Analytical grounds,” PIER 61, 1-26 (2006).
[CrossRef]

A. O. Perov, Y. K. Sirenko, and N. P. Yashina, “Periodic open resonators: peculiarities of pulse scattering and spectral features,” PIER 46, 33-75 (2004).
[CrossRef]

Radio Sci. (1)

Y. K. Sirenko and N. P. Yashina, “Time domain theory of open waveguide resonators: canonical problems and a generalized matrix technique,” Radio Sci. 38, VIC 26-1-VIC 26-12 (2003).
[CrossRef]

Telecommun. Radio Eng. (1)

K. Y. Sirenko, “Splitting of super-broadband pulses by simple inhomogeneities of circular and coaxial waveguide,” Telecommun. Radio Eng. 67, 1415-1428 (2008).
[CrossRef]

Other (8)

V. P. Shestopalov, A. A. Kirilenko, and L. A. Rud', “Waveguide discontinuities,” in Resonance Wave Scattering, Vol. 2 (Naukova Dumka, 1986) (in Russian).

V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, and V. G. Sologub, Wave Diffraction by Gratings (Kharkov State Univ. Press, 1973) (in Russian).

R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

V. P. Shestopalov, A. A. Kirilenko, S. A. Masalov, and Y. K. Sirenko, “Diffraction gratings,” in Resonance Wave Scattering, Vol. 1 (Naukova Dumka, 1986) (in Russian).

V. P. Shestopalov and Y. K. Sirenko, Dynamic Theory of Gratings (Naukova Dumka, 1989) (in Russian).

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).

Y. K. Sirenko, S. Strom, and N. P. Yashina, Modeling and Analysis of Transient Processes in Open Resonant Structures. New Methods and Techniques (Springer, 2007).
[PubMed]

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Springer-Verlag, 1985).

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Figures (11)

Fig. 1
Fig. 1

(a) Echelette geometry and (b) its electrodynamical characteristics in the frequency band 1.4 k 4.2 : H-polarization, α 0 i = 0 ( Φ = 0 ) , l = 4.02 , ψ = 67.5 ° , L 1 = 8.4 .

Fig. 2
Fig. 2

Echelette excitation by quasi-monochromatic pulsed wave (12) with central frequency (a) k ̃ = 1.85 and (b) k ̃ = 2.89 . Parameters H x ( g , t ) and E z ( g , t ) spatial distributions, g Q L at time t = 101 .

Fig. 3
Fig. 3

H-polarization; autocollimation reflection on the minus first spatial harmonic: (a) Grating geometry : L 1 = 8.4 ; 1— l = 4.02 , h = 1.0 , d = 2.02 , ɛ = 1 ; 2— l = 4.02 , h = 1.28 , d = 1.62 , ɛ = 2 ); (b) reflection efficiency: geometry 1, dashed curve; geometry 2, solid curve.

Fig. 4
Fig. 4

Reflective grating excitation (geometry 2) by an H-polarized quasi-monochromatic wave U 0 i ( g , t ) : Φ = 0.5 ; v 0 ( L 1 , t ) = F 2 ( t ) ; k ̃ = 1.565 , T ̃ = 0.5 , T ¯ = 100 : (a) The H x ( g , t ) spatial distribution, g Q L at the time t = 205 ; (b) functions Re U ( g 1 , t ) and U ( τ ) .

Fig. 5
Fig. 5

Autocollimation reflection efficiency on the minus first spatial harmonic: H-polarization, dashed curve; E-polarization, solid curve; l = 2 π , h = 4.3 , d = 3.8 , ɛ = 2 .

Fig. 6
Fig. 6

Autocollimation reflection efficiency on the minus first spatial harmonic: (a) Grating geometry ( l = 4.02 , h = 1.42 , d = 2.78 , ɛ = 2 , perfectly conducting strip thickness is 0.04, L 1 = 8.4 ); (b) reflection efficiency: H-polarization, dashed curve; E-polarization, solid curve.

Fig. 7
Fig. 7

Excitation of a symmetric echelette ( ψ = 45 ° , l = 4.02 , L 1 = 7.8 ) by a normally incident E-polarized quasi-monochromatic wave U 0 i ( g , t ) . The spatial–temporal amplitudes are shown for the U 0 i ( g , t ) wave and the principal spatial harmonics of the secondary field U s ( g , t ) on the virtual boundary L 1 .

Fig. 8
Fig. 8

(Complement to Fig. 7). E x ( g , t ) , H y ( g , t ) and H z ( g , t ) spatial distributions, g Q L , t = 189.75 .

Fig. 9
Fig. 9

Symmetric echelette excitation by a normally incident E-polarized Gaussian pulse U 0 i ( g , t ) : (a) The spatial–temporal and the spectral amplitudes are shown for the U 0 i ( g , t ) wave and the principal spatial harmonics of the secondary field U s ( g , t ) on the virtual boundary L 1 ; (b),(c) the E x ( g , t ) spatial distribution, g Q L , t = 26 (forced oscillations mode) and t = 55 (free oscillations mode).

Fig. 10
Fig. 10

Mode-frequency exfoliation of a superbroadband pulse: (a) Grating geometry ( ψ = 60 ° , l = 4.02 , L 1 = 8.0 ); (b) amplitudes of the incident E-polarized pulsed wave U 0 i ( g , t ) ( Φ = 0 ) ; (c) energy distribution among spatial harmonics of the field U ̃ s ( g , k ) in the structure’s reflection zone.

Fig. 11
Fig. 11

(Complement to Fig. 10). Amplitudes of high-order spatial harmonics of the field U s ( g , t ) on the virtual boundary L 1 .

Equations (24)

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U s ( g , t ) = n = u n ( z , t ) μ n ( y ) and
U i ( g , t ) = n = v n ( z , t ) μ n ( y ) ; t 0 .
{ H y E y } t = η 0 1 U z , { H z E z } t = ± η 0 1 U y ; { E -case H -case } .
{ H y ( z ) s E y ( z ) s } = n = u n y ( z ) ( z , t ) μ n y ( z ) ( y ) ,
{ H y ( z ) i E y ( z ) i } = n = v n y ( z ) ( z , t ) μ n y ( z ) ( ρ ) ; { E -case H -case } .
P j s ( i ) ( t ) = L j ( [ E s ( i ) × H s ( i ) ] n j ) d y j ,
P 1 i × s ( t ) = L 1 [ ( [ E s × H i ] + [ E i × H s ] ) n 1 ] d y 1 ,
f ̃ ( k ) = 0 T f ( t ) e i k t d t f ( t ) ,
R n p A A ( k ) = | u ̃ n ( z 1 , k ) v ̃ p ( z 1 , k ) | z 1 = 0 , T n p B A ( k ) = | u ̃ n ( z 2 , k ) | z 2 = 0 | v ̃ p ( z 1 , k ) | z 1 = 0 .
U ̃ ( g , k ) = { U ̃ p i ( g , k ) + n = n = R n p A A exp [ i ( Φ n y + Γ n ( z L 1 ) ) ] ; g A n = n = T n p B A exp [ i ( Φ n y Γ n ( z + h + L 2 ) ) ] ; g B } ,
n = [ | R n p A A | 2 + | T n p B A | 2 ] { Re Γ n Im Γ n } = { Re Γ p + 2 Im R p p A A Im Γ p Im Γ p 2 Im R p p A A Re Γ p } k 2 β 0 l { W 1 W 2 } .
W 2 = { + } Q L [ μ μ 0 | H | 2 ε ε 0 | E | 2 ] d g ,
β 0 = { η 0 2 η 0 2 } ; { E -case H -case } .
W = W 1 k 2 β 0 l Γ p = k β 0 l Γ p Q L σ ε 0 | E | 2 d g , W n p R = | R n p A A | 2 Re Γ n Γ p ,
W n p T = | T n p B A | 2 Re Γ n Γ p
k l sin ( α p i ) = π ( p + m )
v 0 ( L 1 , t ) = 4 sin [ Δ k ( t T ̃ ) ] ( t T ̃ ) cos [ k ̃ ( t T ̃ ) ] χ ( T ¯ t ) = F 1 ( t ) ; Δ k = 0.7 , T ̃ = 50 , T ¯ = 100 .
U 0 i ( g , t ) : Φ = 0 ; v 0 ( L 1 , t ) = F 1 ( t ) ; k ̃ , Δ k = 0.7 ,
T ̃ = 50 , T ¯ = 100 .
U 0 i ( g , t ) : Φ = 0 ; v 0 ( L 1 , t ) = cos [ k ̃ ( t T ̃ ) ] χ ( T ¯ t ) = F 2 ( t ) ;
k ̃ , T ̃ = 0.5 , T ¯ = 300
W m 0 R ( k ) = 1 , k l sin ( α 0 i ) = π m .
U 0 i ( g , t ) : Φ = 0.5 , v 0 ( L 1 , t ) = F 1 ( t ) , k ̃ = 1.55 ,
Δ k = 0.75 , T ̃ = 50 , T ¯ = 100 .

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