Abstract

The time-domain reflection coefficient for a TM-polarized plane wave obliquely incident on a Lorentz-medium half-space is determined analytically by inversion of the frequency-domain reflection coefficient. The resulting expression contains only the convolution of simple functions. This allows the temporal behavior of the reflection coefficient to be predicted, and the relationship between the material parameters and the oscillation of the response to be easily identified. The time-domain expression is validated numerically through comparison with the inverse fast Fourier transform of the frequency-domain reflection coefficient.

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References

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  1. K. E. Oughstun and G. C. Sherman, "Propagation of electromagnetic pulses in a linear dispersive medium (the Lorentz medium)," J. Opt. Soc. Am. B 5, 817-849 (1988).
    [CrossRef]
  2. K. E. Oughstun and G. C. Sherman, "Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium)," J. Opt. Soc. Am. A 6, 1394-1420 (1989).
    [CrossRef]
  3. K. E. Oughstun and G. C. Sherman, "Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium," Phys. Rev. A 41, 6090-6113 (1990).
    [CrossRef] [PubMed]
  4. E. L. Mokole and S. N. Samaddar, "Transmission and reflection of normally incident, pulsed electromagnetic plane waves upon a Lorentz half-space," J. Opt. Soc. Am. B 16, 812-831 (1999).
    [CrossRef]
  5. J. G. Blaschak and J. Franzen, "Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence," J. Opt. Soc. Am. B 12, 1501-1512 (1995).
    [CrossRef]
  6. G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simpled layered materials," IEEE Trans. Antennas Propag. 51, 3221-3227 (2003).
    [CrossRef]
  7. H.-Y. Pao, S. L. Dvorak, and D. G. Dudley, "An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case)," IEEE Trans. Antennas Propag. 44, 925-932 (1996).
    [CrossRef]
  8. J. C. Oh, E. Rothwell, B. T. Perry, and M. J. Havrilla, "Natural resonance representation of the transient field reflected by a conductor-backed layer of Debye material," J. Electromagn. Waves Appl. 18, 571-589 (2004).
    [CrossRef]
  9. J. W. Suk and E. J. Rothwell, "Transient analysis of TM-plane wave reflection from a layered medium," J. Electromagn. Waves Appl. 16, 1195-1208 (2002).
    [CrossRef]
  10. R. M. Joseph, S. C. Hagness, and A. Taflove, "Direct time integration of Maxwell's equations in linear dispersive media with absorption for scattering and propagation of femtosecond electromagnetic pulses," Opt. Lett. 16, 1412-1414 (1991).
    [CrossRef] [PubMed]
  11. S. M. Cossmann, E. J. Rothwell, and L. C. Kempel, "Transient reflection of TE-polarized plane waves from a Lorentz-medium half-space," J. Opt. Soc. Am. A 23, 2320-2323 (2006).
    [CrossRef]
  12. B. V. Stanic, D. R. Milanovic, and J. M. Cvetic, "Pulse reflection from a lossy Lorentz medium half-space (TM polarization)," J. Phys. D 24, 1245-1249 (1991).
    [CrossRef]
  13. K. G. Gray, "The reflected impulse response of a Lorentz medium," Proc. IEEE 68, 408-409 (1980).
    [CrossRef]
  14. E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2001).
    [CrossRef]
  15. H. A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat (B.G. Teubner, 1909).
  16. W. R. LePage, Complex Variables and the Laplace Transform for Engineers (Dover, 1980).
  17. G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications, 2nd ed. (Van Nostrand, 1951).
  18. M. Abramowitz and I. S. Stegun, Handbook of Mathematical Functions (Dover, 1965).
  19. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

2006 (1)

2004 (1)

J. C. Oh, E. Rothwell, B. T. Perry, and M. J. Havrilla, "Natural resonance representation of the transient field reflected by a conductor-backed layer of Debye material," J. Electromagn. Waves Appl. 18, 571-589 (2004).
[CrossRef]

2003 (1)

G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simpled layered materials," IEEE Trans. Antennas Propag. 51, 3221-3227 (2003).
[CrossRef]

2002 (1)

J. W. Suk and E. J. Rothwell, "Transient analysis of TM-plane wave reflection from a layered medium," J. Electromagn. Waves Appl. 16, 1195-1208 (2002).
[CrossRef]

1999 (1)

1996 (1)

H.-Y. Pao, S. L. Dvorak, and D. G. Dudley, "An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case)," IEEE Trans. Antennas Propag. 44, 925-932 (1996).
[CrossRef]

1995 (1)

J. G. Blaschak and J. Franzen, "Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence," J. Opt. Soc. Am. B 12, 1501-1512 (1995).
[CrossRef]

1991 (2)

1990 (1)

K. E. Oughstun and G. C. Sherman, "Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium," Phys. Rev. A 41, 6090-6113 (1990).
[CrossRef] [PubMed]

1989 (1)

1988 (1)

1980 (1)

K. G. Gray, "The reflected impulse response of a Lorentz medium," Proc. IEEE 68, 408-409 (1980).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. S. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Blaschak, J. G.

J. G. Blaschak and J. Franzen, "Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence," J. Opt. Soc. Am. B 12, 1501-1512 (1995).
[CrossRef]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

Campbell, G. A.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications, 2nd ed. (Van Nostrand, 1951).

Cloud, M. J.

E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2001).
[CrossRef]

Cossmann, S. M.

Cvetic, J. M.

B. V. Stanic, D. R. Milanovic, and J. M. Cvetic, "Pulse reflection from a lossy Lorentz medium half-space (TM polarization)," J. Phys. D 24, 1245-1249 (1991).
[CrossRef]

Dudley, D. G.

H.-Y. Pao, S. L. Dvorak, and D. G. Dudley, "An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case)," IEEE Trans. Antennas Propag. 44, 925-932 (1996).
[CrossRef]

Dvorak, S. L.

H.-Y. Pao, S. L. Dvorak, and D. G. Dudley, "An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case)," IEEE Trans. Antennas Propag. 44, 925-932 (1996).
[CrossRef]

Foster, R. M.

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications, 2nd ed. (Van Nostrand, 1951).

Franzen, J.

J. G. Blaschak and J. Franzen, "Precursor propagation in dispersive media from short-rise-time pulses at oblique incidence," J. Opt. Soc. Am. B 12, 1501-1512 (1995).
[CrossRef]

Frasch, L. L.

G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simpled layered materials," IEEE Trans. Antennas Propag. 51, 3221-3227 (2003).
[CrossRef]

Gray, K. G.

K. G. Gray, "The reflected impulse response of a Lorentz medium," Proc. IEEE 68, 408-409 (1980).
[CrossRef]

Hagness, S. C.

Havrilla, M. J.

J. C. Oh, E. Rothwell, B. T. Perry, and M. J. Havrilla, "Natural resonance representation of the transient field reflected by a conductor-backed layer of Debye material," J. Electromagn. Waves Appl. 18, 571-589 (2004).
[CrossRef]

Joseph, R. M.

Kempel, L. C.

S. M. Cossmann, E. J. Rothwell, and L. C. Kempel, "Transient reflection of TE-polarized plane waves from a Lorentz-medium half-space," J. Opt. Soc. Am. A 23, 2320-2323 (2006).
[CrossRef]

G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simpled layered materials," IEEE Trans. Antennas Propag. 51, 3221-3227 (2003).
[CrossRef]

LePage, W. R.

W. R. LePage, Complex Variables and the Laplace Transform for Engineers (Dover, 1980).

Lorentz, H. A.

H. A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat (B.G. Teubner, 1909).

Milanovic, D. R.

B. V. Stanic, D. R. Milanovic, and J. M. Cvetic, "Pulse reflection from a lossy Lorentz medium half-space (TM polarization)," J. Phys. D 24, 1245-1249 (1991).
[CrossRef]

Mokole, E. L.

Nyquist, D. P.

G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simpled layered materials," IEEE Trans. Antennas Propag. 51, 3221-3227 (2003).
[CrossRef]

Oh, J. C.

J. C. Oh, E. Rothwell, B. T. Perry, and M. J. Havrilla, "Natural resonance representation of the transient field reflected by a conductor-backed layer of Debye material," J. Electromagn. Waves Appl. 18, 571-589 (2004).
[CrossRef]

Oughstun, K. E.

Pao, H.-Y.

H.-Y. Pao, S. L. Dvorak, and D. G. Dudley, "An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case)," IEEE Trans. Antennas Propag. 44, 925-932 (1996).
[CrossRef]

Perry, B. T.

J. C. Oh, E. Rothwell, B. T. Perry, and M. J. Havrilla, "Natural resonance representation of the transient field reflected by a conductor-backed layer of Debye material," J. Electromagn. Waves Appl. 18, 571-589 (2004).
[CrossRef]

Rothwell, E.

J. C. Oh, E. Rothwell, B. T. Perry, and M. J. Havrilla, "Natural resonance representation of the transient field reflected by a conductor-backed layer of Debye material," J. Electromagn. Waves Appl. 18, 571-589 (2004).
[CrossRef]

Rothwell, E. J.

S. M. Cossmann, E. J. Rothwell, and L. C. Kempel, "Transient reflection of TE-polarized plane waves from a Lorentz-medium half-space," J. Opt. Soc. Am. A 23, 2320-2323 (2006).
[CrossRef]

G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simpled layered materials," IEEE Trans. Antennas Propag. 51, 3221-3227 (2003).
[CrossRef]

J. W. Suk and E. J. Rothwell, "Transient analysis of TM-plane wave reflection from a layered medium," J. Electromagn. Waves Appl. 16, 1195-1208 (2002).
[CrossRef]

E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2001).
[CrossRef]

Samaddar, S. N.

Sherman, G. C.

Stanic, B. V.

B. V. Stanic, D. R. Milanovic, and J. M. Cvetic, "Pulse reflection from a lossy Lorentz medium half-space (TM polarization)," J. Phys. D 24, 1245-1249 (1991).
[CrossRef]

Stegun, I. S.

M. Abramowitz and I. S. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Stenholm, G. J.

G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simpled layered materials," IEEE Trans. Antennas Propag. 51, 3221-3227 (2003).
[CrossRef]

Suk, J. W.

J. W. Suk and E. J. Rothwell, "Transient analysis of TM-plane wave reflection from a layered medium," J. Electromagn. Waves Appl. 16, 1195-1208 (2002).
[CrossRef]

Taflove, A.

IEEE Trans. Antennas Propag. (2)

G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simpled layered materials," IEEE Trans. Antennas Propag. 51, 3221-3227 (2003).
[CrossRef]

H.-Y. Pao, S. L. Dvorak, and D. G. Dudley, "An accurate and efficient analysis for transient plane waves obliquely incident on a conductive half space (TM case)," IEEE Trans. Antennas Propag. 44, 925-932 (1996).
[CrossRef]

J. Electromagn. Waves Appl. (2)

J. C. Oh, E. Rothwell, B. T. Perry, and M. J. Havrilla, "Natural resonance representation of the transient field reflected by a conductor-backed layer of Debye material," J. Electromagn. Waves Appl. 18, 571-589 (2004).
[CrossRef]

J. W. Suk and E. J. Rothwell, "Transient analysis of TM-plane wave reflection from a layered medium," J. Electromagn. Waves Appl. 16, 1195-1208 (2002).
[CrossRef]

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

J. Opt. Soc. Am. B (3)

J. Phys. D (1)

B. V. Stanic, D. R. Milanovic, and J. M. Cvetic, "Pulse reflection from a lossy Lorentz medium half-space (TM polarization)," J. Phys. D 24, 1245-1249 (1991).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

K. E. Oughstun and G. C. Sherman, "Uniform asymptotic description of ultrashort rectangular optical pulse propagation in a linear, causally dispersive medium," Phys. Rev. A 41, 6090-6113 (1990).
[CrossRef] [PubMed]

Proc. IEEE (1)

K. G. Gray, "The reflected impulse response of a Lorentz medium," Proc. IEEE 68, 408-409 (1980).
[CrossRef]

Other (6)

E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2001).
[CrossRef]

H. A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat (B.G. Teubner, 1909).

W. R. LePage, Complex Variables and the Laplace Transform for Engineers (Dover, 1980).

G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications, 2nd ed. (Van Nostrand, 1951).

M. Abramowitz and I. S. Stegun, Handbook of Mathematical Functions (Dover, 1965).

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

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Figures (5)

Fig. 1
Fig. 1

Time-domain reflection coefficient with incidence angle θ = 30 ° and material parameters ω 0 = 4.0 × 10 16 s 1 , b 2 = 20.0 × 10 32 s 2 , and δ = 0.28 × 10 16 s 1 .

Fig. 2
Fig. 2

Time-domain reflection coefficient with incidence angle θ = 30 ° and material parameters ω 0 = 2.0 × 10 15 s 1 , b 2 = 20.0 × 10 29 s 2 , δ = 0.28 × 10 16 s 1 .

Fig. 3
Fig. 3

Time-domain reflection coefficient with incidence angle θ = 30 ° with parameter choices of ω 0 = 2.0 × 10 15 s 1 , b 2 = 20.0 × 10 32 s 2 , δ = 0.28 × 10 16 s 1 .

Fig. 4
Fig. 4

Time-domain reflection coefficient with incidence angle θ = 50 ° and material parameters ω 0 = 4.0 × 10 16 s 1 , b 2 = 20.0 × 10 32 s 2 , and δ = 0.28 × 10 16 s 1 . This case results in a noncausal behavior for c ( t ) .

Fig. 5
Fig. 5

Time-domain reflection coefficient with incidence angle θ = 45 ° and material parameters ω 0 = 4.0 × 10 16 s 1 , b 2 = 20.0 × 10 32 s 2 , δ = 0.28 × 10 16 s 1 .

Equations (52)

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Γ ( ω ) = Z ( ω ) Z 0 Z ( ω ) + Z 0 .
Z ( ω ) = η ( ω ) k z ( ω ) k ( ω ) ,
ϵ r ( ω ) = ϵ ω 0 2 ( ϵ s ϵ ) ω 2 2 j ω δ ω 0 2 .
ϵ r ( ω ) = 1 + b 2 ω 0 2 ω 2 + 2 j δ ω .
Γ ( s ) = [ ( s s 1 ) ( s s 2 ) ( s s 5 ) ( s s 6 ) ] 1 2 ( s s 3 ) ( s s 4 ) [ ( s s 1 ) ( s s 2 ) ( s s 5 ) ( s s 6 ) ] 1 2 + ( s s 3 ) ( s s 4 ) = N ( s ) D ( s ) ,
s 1 , 2 = δ ± λ 1 , λ 1 = ( δ 2 ω 0 2 ) 1 2 ,
s 3 , 4 = δ ± λ 3 , λ 3 = ( δ 2 ω 0 2 b 2 ) 1 2 ,
s 5 , 6 = δ ± λ 5 , λ 5 = ( δ 2 ω 0 2 B 2 ) 1 2 ,
Γ ( s ) = N ( s ) D ( s ) N ( s ) N ( s ) .
D ( s ) N ( s ) = b 2 ( tan 2 θ 1 ) ( s s A ) ( s s B ) ,
s A , B = δ ± λ A , λ A = [ δ 2 ω 0 2 b 2 ( tan 2 θ 1 ) ] 1 2 .
[ N ( s ) ] 2 = ( s s 1 ) ( s s 2 ) G ( s ) ,
G ( s ) = g 1 ( s ) + g 2 ( s ) 2 b 2 g 3 ( s ) + b 4 g 4 ( s ) + 2 b 2 ,
g 1 ( s ) = ( s s 1 ) ( s s 2 ) [ ( s s 1 ) ( s s 2 ) ( s s 5 ) ( s s 6 ) ] 1 2 ,
g 2 ( s ) = ( s s 5 ) ( s s 6 ) [ ( s s 1 ) ( s s 2 ) ( s s 5 ) ( s s 6 ) ] 1 2 ,
g 3 ( s ) = ( s s 5 ) ( s s 6 ) [ ( s s 1 ) ( s s 2 ) ( s s 5 ) ( s s 6 ) ] 1 2 ,
g 4 ( s ) = 1 ( s s 1 ) ( s s 2 ) .
b 2 ( tan 2 θ 1 ) Γ ( t ) = c ( t ) g ( t ) ,
g ( t ) = g 1 ( t ) + g 2 ( t ) 2 b 2 g 3 ( t ) + b 4 g 4 ( t ) + 2 b 2 δ ( t ) ,
C ( s ) = ( s s 1 ) ( s s 2 ) ( s s A ) ( s s B ) .
g 1 ( t ) + g 2 ( t ) = 2 exp ( δ t ) { λ 1 2 λ 5 2 [ I ̂ 1 ( λ 1 t ) u ( t ) ] [ I ̂ 1 ( λ 5 t ) u ( t ) ] + λ 1 3 I ̂ 2 ( λ 1 t ) u ( t ) + λ 5 3 I ̂ 2 ( λ 5 t ) u ( t ) } ,
I ̂ n ( x ) = I n ( x ) x ,
1 ( s + ρ ) 1 2 ( s + σ ) 1 2 exp ( [ ρ + σ ] t 2 ) I 0 ( [ ρ σ ] t 2 ) u ( t )
g 3 ( t ) = ( d 2 d t 2 + 2 δ d d t + ω 0 2 + b 2 cos 2 θ ) g ¯ 3 ( t ) ,
g ¯ 3 ( t ) = exp ( δ t ) [ { I 0 ( λ 1 t ) u ( t ) } { I 0 ( λ 5 t ) u ( t ) } ] .
g 3 ( t ) = ( d 2 d t 2 λ 5 2 ) g ¯ 3 ( t ) .
d d t [ f ( t ) g ( t ) ] = d f ( t ) d t g ( t ) = f ( t ) d g ( t ) d t ,
g 3 ( t ) = exp ( δ t ) [ λ 5 2 { I 0 ( λ 1 t ) u ( t ) } { I ̂ 1 ( λ 5 t ) u ( t ) } + λ 1 I 1 ( λ 1 t ) u ( t ) ] + δ ( t ) .
g 4 ( s ) = 1 ( s s 1 ) ( s s 2 ) = 1 2 λ 1 ( 1 s s 1 1 s s 2 ) ,
g 4 ( t ) = exp ( δ t ) λ 1 sinh ( λ 1 t ) u ( t ) .
g ( t ) = 2 exp ( δ t ) [ λ 1 2 λ 5 2 { I ̂ 1 ( λ 1 t ) u ( t ) } { I ̂ 1 ( λ 5 t ) u ( t ) } + λ 1 3 I ̂ 2 ( λ 1 t ) u ( t ) + λ 5 3 I ̂ 2 ( λ 5 t ) u ( t ) + b 2 λ 5 2 { I 0 ( λ 1 t ) u ( t ) } { I ̂ 1 ( λ 5 t ) u ( t ) } b 2 λ 1 I 1 ( λ 1 t ) u ( t ) + b 4 2 λ 1 sinh ( λ 1 t ) u ( t ) ] .
λ ¯ 1 = ( ω 0 2 δ 2 ) 1 2 , λ ¯ 5 = ( ω 0 2 + B 2 δ 2 ) 1 2
g ( t ) = 2 exp ( δ t ) [ λ ¯ 1 2 λ ¯ 5 2 { J ̂ 1 ( λ ¯ 1 t ) u ( t ) } { J ̂ 1 ( λ ¯ 5 t ) u ( t ) } + λ ¯ 1 3 J ̂ 2 ( λ ¯ 1 t ) u ( t ) + λ ¯ 5 3 J ̂ 2 ( λ ¯ 5 t ) u ( t ) b 2 λ ¯ 5 2 { J 0 ( λ ¯ 1 t ) u ( t ) } { J ̂ 1 ( λ ¯ 5 t ) u ( t ) } + b 2 λ ¯ 1 J 1 ( λ ¯ 1 t ) u ( t ) + b 4 2 λ ¯ 1 sin ( λ ¯ 1 t ) u ( t ) ] .
g ( t ) = 2 exp ( δ t ) [ λ 1 2 λ ¯ 5 2 { I ̂ 1 ( λ 1 t ) u ( t ) } { J ̂ 1 ( λ ¯ 5 t ) u ( t ) } + λ 1 3 I ̂ 2 ( λ 1 t ) u ( t ) + λ ¯ 5 3 J ̂ 2 ( λ ¯ 5 t ) u ( t ) + b 2 λ ¯ 5 2 { I 0 ( λ 1 t ) u ( t ) } { J ̂ 1 ( λ ¯ 5 t ) u ( t ) } b 2 λ 1 I 1 ( λ 1 t ) u ( t ) + b 4 2 λ 1 sinh ( λ 1 t ) u ( t ) ] .
C ( s ) = 1 + b 2 2 λ A ( tan 2 θ 1 ) [ 1 s s A 1 s s B ] .
c ( t ) = δ ( t ) b 2 exp ( δ t ) λ A ( tan 2 θ 1 ) sinh ( λ A t ) u ( t ) .
λ ¯ A = ( ω 0 2 δ 2 + b 2 tan 2 θ 1 ) 1 2 ,
c ( t ) = δ ( t ) + b 2 exp ( δ t ) λ ¯ A ( tan 2 θ 1 ) sin ( λ ¯ A t ) u ( t ) .
c ( t ) = δ ( t ) b 2 exp ( δ t ) λ A ( tan 2 θ 1 ) [ exp ( λ A t ) u ( t ) exp ( λ A t ) u ( t ) ] = δ ( t ) b 2 exp ( δ t ) λ A ( tan 2 θ 1 ) exp ( λ A t ) .
b 4 Γ ( s ) = ( s s 1 ) ( s s 2 ) G ( s ) = ( s 2 + 2 δ s + ω 0 ) G ( s ) .
b 4 Γ ( t ) = ( d 2 d t 2 + 2 δ d d t + ω 0 2 ) g ( t ) ,
b 4 Γ ( t ) = 2 exp ( δ t ) ( d 2 d t 2 λ 1 2 ) g A ( t ) ,
g A ( t ) = λ 1 3 λ 5 3 { I ̂ 1 ( λ 1 t ) u ( t ) } { I ̂ 1 ( λ 5 t ) u ( t ) } 1 2 λ 1 3 λ 5 2 I ̂ 1 ( λ 1 t ) u ( t ) 1 2 λ 5 5 I ̂ 1 ( λ 5 t ) u ( t ) + λ 1 5 I ̂ 2 ( λ 1 t ) u ( t ) + λ 5 5 I ̂ 2 ( λ 5 t ) u ( t ) + b 2 λ 1 2 λ 5 2 { I 1 ( λ 1 t ) u ( t ) } { I ̂ 1 ( λ 5 t ) u ( t ) } b 2 λ 1 3 I 1 ( λ 1 t ) u ( t ) + 1 2 b 4 λ 1 sinh ( λ 1 t ) u ( t ) ,
I ̂ n ( x ) = n 1 2 n I ̂ n 1 ( x ) + n + 1 2 n I ̂ n + 1 ( x ) ,
Γ ( t ) = 2 exp ( δ t ) b 4 [ λ 1 3 λ 5 3 { I ̂ 2 ( λ 1 t ) u ( t ) } { I ̂ 2 ( λ 5 t ) u ( t ) } + 1 2 λ 1 2 λ 5 2 ( λ 1 2 + λ 5 2 ) { I ̂ 1 ( λ 1 t ) u ( t ) } { I ̂ 1 ( λ 5 t ) u ( t ) } λ 1 3 λ 5 2 I ̂ 2 ( λ 1 t ) u ( t ) λ 1 2 λ 5 3 I ̂ 2 ( λ 5 t ) u ( t ) + 1 2 λ 1 5 I ̂ 4 ( λ 1 t ) u ( t ) + 1 2 λ 5 5 I ̂ 4 ( λ 5 t ) u ( t ) ] .
Γ ( t ) = 2 exp ( δ t ) b 4 [ λ ¯ 1 3 λ ¯ 5 3 { J ̂ 2 ( λ ¯ 1 t ) u ( t ) } { J ̂ 2 ( λ ¯ 5 t ) u ( t ) } 1 2 λ ¯ 1 2 λ ¯ 5 2 ( λ ¯ 1 2 + λ ¯ 5 2 ) { J ̂ 1 ( λ ¯ 1 t ) u ( t ) } { J ̂ 1 ( λ ¯ 5 t ) u ( t ) } + λ ¯ 1 3 λ ¯ 5 2 J ̂ 2 ( λ ¯ 1 t ) u ( t ) + λ ¯ 1 2 λ ¯ 5 3 J ̂ 2 ( λ ¯ 5 t ) u ( t ) + 1 2 λ ¯ 1 5 J ̂ 4 ( λ ¯ 1 t ) u ( t ) + 1 2 λ ¯ 5 5 J ̂ 4 ( λ ¯ 5 t ) u ( t ) ] .
H ( s ) = [ s + β s + β ] F ( s ) = ( 1 s + β ) [ ( s + β ) F ( s ) ] ,
h ( t ) = [ exp ( β t ) u ( t ) ] [ f ( t ) + β f ( t ) ] = h 1 ( t ) + h 2 ( t ) ,
h 1 ( t ) = 0 exp ( β [ t τ ] ) f ( τ ) d τ = exp ( β t ) 0 exp ( β τ ) f ( τ ) d τ .
h 1 ( t ) = exp ( β t ) [ exp ( β τ ) f ( τ ) 0 β 0 exp ( β τ ) f ( τ ) d τ ]
= β exp ( β t ) 0 exp ( β τ ) f ( τ ) d τ .
h 2 ( t ) = β 0 exp ( β [ t τ ] ) f ( τ ) d τ = β exp ( β t ) 0 exp ( β τ ) f ( τ ) d τ .

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