Abstract

The time-domain reflection coefficient for a 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 simple functions and a single convolution of these functions. Owing to its simplicity, this form of the reflection coefficient provides insight into its temporal behavior, specifically how the relationship between the damping coefficient and the oscillation frequency determines the shape of the response. The simple form of the reflection coefficient 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. 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]
  4. 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-l512 (1995).
    [CrossRef]
  5. G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simple layered materials," IEEE Trans. Antennas Propag. 51, 3221-3227 (2003).
    [CrossRef]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. J. A. Marozas and K. E. Oughstun, "Electromagnetic pulse propagation across a planar interface separating two lossy, dispersive dielectrics," in Ultra-Wideband, Short-Pulse Electromagnetics 3 (Plenum, 1997), pp. 217-230.
  11. K. G. Gray, "The reflected impulse response of a Lorentz medium," Proc. IEEE 68, 408-409 (1980).
    [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. E. J. Rothwell and M. J. Cloud, Electromagnetics (CRC Press, 2001).
    [CrossRef]
  14. W. R. LePage, Complex Variables and the Laplace Transform for Engineers (Dover, 1980).
  15. G. A. Campbell and R. M. Foster, Fourier Integrals for Practical Applications, 2nd ed. (Van Nostrand, 1951).
  16. M. Abramowitz and I. S. Stegun, Handbook of Mathematical Functions (Dover, 1965).
  17. Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, 1962).
  18. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

2004

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

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

2002

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]

2001

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

1999

1997

J. A. Marozas and K. E. Oughstun, "Electromagnetic pulse propagation across a planar interface separating two lossy, dispersive dielectrics," in Ultra-Wideband, Short-Pulse Electromagnetics 3 (Plenum, 1997), pp. 217-230.

1996

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

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-l512 (1995).
[CrossRef]

1991

1989

1988

1980

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

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

1965

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

1962

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, 1962).

1960

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

1951

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

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-l512 (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]

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-l512 (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 simple 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.

G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simple 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).

Luke, Y. L.

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, 1962).

Marozas, J. A.

J. A. Marozas and K. E. Oughstun, "Electromagnetic pulse propagation across a planar interface separating two lossy, dispersive dielectrics," in Ultra-Wideband, Short-Pulse Electromagnetics 3 (Plenum, 1997), pp. 217-230.

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 simple 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.

G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simple 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 simple 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.

G. J. Stenholm, E. J. Rothwell, D. P. Nyquist, L. C. Kempel, and L. L. Frasch, "E-pulse diagnostics of simple 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.

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

J. Opt. Soc. Am. B

J. Phys. D

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.

Proc. IEEE

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

Other

J. A. Marozas and K. E. Oughstun, "Electromagnetic pulse propagation across a planar interface separating two lossy, dispersive dielectrics," in Ultra-Wideband, Short-Pulse Electromagnetics 3 (Plenum, 1997), pp. 217-230.

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

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).

Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, 1962).

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

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Figures (3)

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 , δ = 0.28 × 10 16 s 1 . This choice of parameters corresponds to case 1, Eq. (37).

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 . The choice of parameters corresponds to case 2, Eq. (34).

Fig. 3
Fig. 3

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 32 s 2 , δ = 0.28 × 10 16 s 1 . The choice of parameters corresponds to case 3, Eq. (39).

Equations (39)

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Γ ( ω ) = Z ( ω ) Z 0 Z ( ω ) + Z 0 ,
Z ( ω ) = η ( ω ) k ( ω ) k z ( 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 2 + 2 δ s + ω 0 2 ) 1 2 ( s 2 + 2 δ s + ω 0 2 + B 2 ) 1 2 ( s 2 + 2 δ s + ω 0 2 ) 1 2 + ( s 2 + 2 δ s + ω 0 2 + B 2 ) 1 2 ,
Γ ( s ) = [ ( s s 1 ) ( s s 2 ) ] 1 2 [ ( s s 3 ) ( s s 4 ) ] 1 2 [ ( s s 1 ) ( s s 2 ) ] 1 2 + [ ( s s 3 ) ( s s 4 ) ] 1 2 ,
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 .
Γ ( s ) = [ F ( s ) ] 2 B 2 ,
F ( s ) = [ ( s s 1 ) ( s s 2 ) ] 1 2 [ ( s s 3 ) ( s s 4 ) ] 1 2 .
F ( s ) = s { [ ( s s 2 s s 1 ) 1 2 1 ] [ ( s s 4 s s 3 ) 1 2 1 ] } s 1 [ ( s s 2 s s 2 ) 1 2 1 ] + s 3 [ ( s s 4 s s 3 ) 1 2 1 ] ( s 1 s 3 ) .
( s s 2 s s 1 ) 1 2 1 f 1 ( t ) ,
( s s 4 s s 3 ) 1 2 1 f 2 ( t ) .
L 1 { F ( s ) } = f ( t )
= d d t [ f 1 ( t ) f 2 ( t ) ] s 1 f 1 ( t ) + s 3 f 2 ( t ) ( s 1 s 3 ) δ ( t ) .
( s α s β ) 1 2 1 1 2 ( α + β ) exp [ 1 2 ( α β ) t ] { I 1 [ 1 2 ( α + β ) t ] + I 0 [ 1 2 ( α + β ) t ] } u ( t )
f 1 ( t ) = λ 1 exp ( δ t ) [ I 1 ( λ 1 t ) + I 0 ( λ 1 t ) ] u ( t ) ,
f 2 ( t ) = λ 3 exp ( δ t ) [ I 1 ( λ 3 t ) + I 0 ( λ 3 t ) ] u ( t ) .
d f 1 ( t ) d t = λ 1 exp ( δ t ) [ λ 1 I 1 ( λ 1 t ) + ( λ 1 δ ) I 1 ( λ 1 t ) δ I 0 ( λ 1 t ) ] u ( t ) + λ 1 δ ( t ) ,
d f 2 ( t ) d t = λ 3 exp ( δ t ) [ λ 3 I 1 ( λ 3 t ) + ( λ 3 δ ) I 1 ( λ 3 t ) δ I 0 ( λ 3 t ) ] u ( t ) + λ 3 δ ( t ) .
f ( t ) = λ 1 2 exp ( δ t ) [ I 1 ( λ 1 t ) I 0 ( λ 1 t ) ] u ( t ) λ 3 2 exp ( δ t ) [ I 1 ( λ 3 t ) I 0 ( λ 3 t ) ] u ( t ) .
1 x I 1 ( x ) = I 1 ( x ) + I 0 ( x )
f ( t ) = [ λ 1 2 exp ( δ t ) I 1 ( λ 1 t ) λ 1 t + λ 3 2 exp ( δ t ) I 1 ( λ 3 t ) λ 3 t ] u ( t ) .
B 2 Γ ( t ) = f ( t ) * f ( t )
= exp ( δ t ) ( { [ λ 1 2 I 1 ( λ 1 t ) λ 1 t + λ 3 2 I 1 ( λ 3 t ) λ 3 t ] u ( t ) } * { [ λ 1 2 I 1 ( λ 1 t ) λ 1 t + λ 3 2 I 1 ( λ 3 t ) λ 3 t ] u ( t ) } )
= λ 1 4 exp ( δ t ) { [ I 1 ( λ 1 t ) λ 1 t u ( t ) ] * [ I 1 ( λ 1 t ) λ 1 t u ( t ) ] } 2 λ 1 2 λ 3 2 exp ( δ t ) { [ I 1 ( λ 1 t ) λ 1 t u ( t ) ] * [ I 1 ( λ 3 t ) λ 3 t u ( t ) ] } + λ 3 4 exp ( δ t ) { [ I 1 ( λ 3 t ) λ 3 t u ( t ) ] * [ I 1 ( λ 3 t ) λ 3 t u ( t ) ] }
= λ 1 4 exp ( δ t ) f A ( t ) 2 λ 1 2 λ 3 2 exp ( δ t ) f B ( t ) + λ 3 4 exp ( δ t ) f C ( t ) .
f A ( t ) = u ( t ) 0 t I 1 ( λ 1 t ) λ 1 t I 1 ( λ 1 [ t t ] ) λ 1 [ t t ] d t .
f A ( t ) = u ( t ) λ 1 t 0 I 1 ( λ 1 t u ) λ 1 t u I 1 ( u ) u ( 1 λ 1 d u ) .
f A ( t ) = u ( t ) λ 1 0 τ I 1 ( τ u ) τ u I 1 ( u ) u d u
= u ( t ) λ 1 [ 2 τ I 2 ( τ ) ] .
f A ( t ) = 2 λ 1 I ̂ 2 ( λ 1 t ) , f C ( t ) = 2 λ 3 I ̂ 2 ( λ 3 t ) ,
I ̂ n ( x ) = I n ( x ) x u ( x ) .
Γ ( t ) = 2 B 2 exp ( δ t ) [ λ 1 3 I ̂ 2 ( λ 1 t ) + λ 3 3 I ̂ 2 ( λ 3 t ) λ 1 2 λ 3 2 I ̂ 1 ( λ 1 t ) * I ̂ 1 ( λ 3 t ) ] .
λ 1 ¯ = ( ω 0 2 δ 2 ) 1 2 , λ 3 ¯ = ( ω 0 2 + B 2 δ 2 ) 1 2
I n ( j x ) = j n J n ( x )
Γ ( t ) = 2 B 2 exp ( δ t ) [ λ ¯ 1 3 J ̂ 2 ( λ ¯ 1 t ) + λ ¯ 3 3 J ̂ 2 ( λ ¯ 3 t ) λ ¯ 1 2 λ ¯ 3 2 J ̂ 1 ( λ ¯ 1 t ) * J ̂ 1 ( λ ¯ 3 t ) ] ,
J ̂ n ( x ) = J n ( x ) x u ( x ) .
Γ ( t ) = 2 B 2 exp ( δ t ) [ λ 1 3 I ̂ 2 ( λ 1 t ) + λ ¯ 3 3 J ̂ 2 ( λ ¯ 3 t ) + λ 1 2 λ ¯ 3 2 I ̂ 1 ( λ 1 t ) * J ̂ 1 ( λ ¯ 3 t ) ] .

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