Abstract

Numerical research is reported on the propagation of short microwave pulses into living, biological materials. These materials are dispersive, and data on the dielectric constant and conductivity for these materials follow a Debye model. A Fourier-series calculation is presented that predicts the occurrence of Brillouin precursors when the incident pulses have sufficiently short rise times. These transients are attenuated with increasing propagation distance but are attenuated more slowly than the carrier frequency of the pulse, which is attenuated exponentially with distance. An analysis of the numerical error resulting from truncation of the Fourier series is given. Upper-bound estimates of truncation error show good series convergence.

© 1989 Optical Society of America

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References

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  1. J. Benford, D. Price, H. Sze, D. Bromley, “Interaction of a vircator microwave generator with an enclosing resonant cavity,” J. Appl. Phys. 61, 2098–2100 (1987).
    [Crossref]
  2. H. Sze, J. Benford, W. Woo, B. Harteneck, “Dynamics of a virtual cathode oscillator driven by a pinched diode,” Phys. Fluids 29, 3873–3880 (1986).
    [Crossref]
  3. J. Benford, H. Sze, T. Young, D. Bromley, G. Prouix, “Variations on the relativistic magnetron,”IEEE Trans. Plasma Sci. PS-13, 538–544 (1985).
    [Crossref]
  4. E. H. Grant, King’s College, University of London, London, UK (personal communication, 1988). See also E. H. Grant, R. J. Sheppard, G. P. South, Dielectric Behaviour of Biological Molecules in Solution (Clarendon, Oxford, 1978).
  5. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  6. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
    [Crossref]
  7. L. Brillouin, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
    [Crossref]
  8. G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
    [Crossref]
  9. K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
    [Crossref]

1988 (1)

1987 (1)

J. Benford, D. Price, H. Sze, D. Bromley, “Interaction of a vircator microwave generator with an enclosing resonant cavity,” J. Appl. Phys. 61, 2098–2100 (1987).
[Crossref]

1986 (1)

H. Sze, J. Benford, W. Woo, B. Harteneck, “Dynamics of a virtual cathode oscillator driven by a pinched diode,” Phys. Fluids 29, 3873–3880 (1986).
[Crossref]

1985 (1)

J. Benford, H. Sze, T. Young, D. Bromley, G. Prouix, “Variations on the relativistic magnetron,”IEEE Trans. Plasma Sci. PS-13, 538–544 (1985).
[Crossref]

1981 (1)

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[Crossref]

1914 (2)

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[Crossref]

L. Brillouin, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

Benford, J.

J. Benford, D. Price, H. Sze, D. Bromley, “Interaction of a vircator microwave generator with an enclosing resonant cavity,” J. Appl. Phys. 61, 2098–2100 (1987).
[Crossref]

H. Sze, J. Benford, W. Woo, B. Harteneck, “Dynamics of a virtual cathode oscillator driven by a pinched diode,” Phys. Fluids 29, 3873–3880 (1986).
[Crossref]

J. Benford, H. Sze, T. Young, D. Bromley, G. Prouix, “Variations on the relativistic magnetron,”IEEE Trans. Plasma Sci. PS-13, 538–544 (1985).
[Crossref]

Brillouin, L.

L. Brillouin, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

Bromley, D.

J. Benford, D. Price, H. Sze, D. Bromley, “Interaction of a vircator microwave generator with an enclosing resonant cavity,” J. Appl. Phys. 61, 2098–2100 (1987).
[Crossref]

J. Benford, H. Sze, T. Young, D. Bromley, G. Prouix, “Variations on the relativistic magnetron,”IEEE Trans. Plasma Sci. PS-13, 538–544 (1985).
[Crossref]

Grant, E. H.

E. H. Grant, King’s College, University of London, London, UK (personal communication, 1988). See also E. H. Grant, R. J. Sheppard, G. P. South, Dielectric Behaviour of Biological Molecules in Solution (Clarendon, Oxford, 1978).

Harteneck, B.

H. Sze, J. Benford, W. Woo, B. Harteneck, “Dynamics of a virtual cathode oscillator driven by a pinched diode,” Phys. Fluids 29, 3873–3880 (1986).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Oughstun, K. E.

K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[Crossref]

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[Crossref]

Price, D.

J. Benford, D. Price, H. Sze, D. Bromley, “Interaction of a vircator microwave generator with an enclosing resonant cavity,” J. Appl. Phys. 61, 2098–2100 (1987).
[Crossref]

Prouix, G.

J. Benford, H. Sze, T. Young, D. Bromley, G. Prouix, “Variations on the relativistic magnetron,”IEEE Trans. Plasma Sci. PS-13, 538–544 (1985).
[Crossref]

Sherman, G. C.

K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
[Crossref]

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[Crossref]

Sommerfeld, A.

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[Crossref]

Sze, H.

J. Benford, D. Price, H. Sze, D. Bromley, “Interaction of a vircator microwave generator with an enclosing resonant cavity,” J. Appl. Phys. 61, 2098–2100 (1987).
[Crossref]

H. Sze, J. Benford, W. Woo, B. Harteneck, “Dynamics of a virtual cathode oscillator driven by a pinched diode,” Phys. Fluids 29, 3873–3880 (1986).
[Crossref]

J. Benford, H. Sze, T. Young, D. Bromley, G. Prouix, “Variations on the relativistic magnetron,”IEEE Trans. Plasma Sci. PS-13, 538–544 (1985).
[Crossref]

Woo, W.

H. Sze, J. Benford, W. Woo, B. Harteneck, “Dynamics of a virtual cathode oscillator driven by a pinched diode,” Phys. Fluids 29, 3873–3880 (1986).
[Crossref]

Young, T.

J. Benford, H. Sze, T. Young, D. Bromley, G. Prouix, “Variations on the relativistic magnetron,”IEEE Trans. Plasma Sci. PS-13, 538–544 (1985).
[Crossref]

Ann. Phys. (2)

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[Crossref]

L. Brillouin, “Über die Fortpflanzung des Lichtes in disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

IEEE Trans. Plasma Sci. (1)

J. Benford, H. Sze, T. Young, D. Bromley, G. Prouix, “Variations on the relativistic magnetron,”IEEE Trans. Plasma Sci. PS-13, 538–544 (1985).
[Crossref]

J. Appl. Phys. (1)

J. Benford, D. Price, H. Sze, D. Bromley, “Interaction of a vircator microwave generator with an enclosing resonant cavity,” J. Appl. Phys. 61, 2098–2100 (1987).
[Crossref]

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

Phys. Fluids (1)

H. Sze, J. Benford, W. Woo, B. Harteneck, “Dynamics of a virtual cathode oscillator driven by a pinched diode,” Phys. Fluids 29, 3873–3880 (1986).
[Crossref]

Phys. Rev. Lett. (1)

G. C. Sherman, K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett. 47, 1451–1454 (1981).
[Crossref]

Other (2)

E. H. Grant, King’s College, University of London, London, UK (personal communication, 1988). See also E. H. Grant, R. J. Sheppard, G. P. South, Dielectric Behaviour of Biological Molecules in Solution (Clarendon, Oxford, 1978).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

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Figures (7)

Fig. 1
Fig. 1

Relative permittivity or dielectric constant (dashed line) and conductance (in reciprocal ohms per meter), shown as functions of frequency. Logarithmic scales are used.

Fig. 2
Fig. 2

Example trapezoidal pulse. The trapezoidal envelope can be modified to achieve the extremes of square-wave or triangular-wave modulation.

Fig. 3
Fig. 3

Diagram of the half-space geometry studied in this paper.

Fig. 4
Fig. 4

Transmitted wave at depths of (a) 5 cm, (b) 75 cm, and (c) 150 cm in a water half-space when the incident wave is a square-wave-modulated, 1-V/m, 1-GHz sinusoidal field orthogonally incident upon the air-half-space interface.

Fig. 5
Fig. 5

Peak of the leading transient (dashed curve) and amplitude of the fundamental frequency (solid-curve), as functions of the propagation depth in the half-space z > 0 for the 1-GHz, square-wave-modulated signal.

Fig. 6
Fig. 6

Transmitted wave at a depth of 150 cm in a water half-space for an incident wave that is a trapezoidally modulated, 1-V/m, 1-GHz sinusoid. The rise-time parameter a, the time to reach the maximum value of the trapezoidal modulation, is set at (a) 1/(2f) sec, (b) 3/(4f) sec, and (c) 3/(2f) sec.

Fig. 7
Fig. 7

Peak of the leading transient (dashed curve) and the amplitude of the fundamental frequency (solid curve), as functions of the propagation depth in the half-space z > 0 for the 10-GHz, square-wave-modulated signal.

Equations (23)

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r = 5.5 + 72.7 / [ 1 + ( ω T ) 2 ] , σ = 10 - 5 + 72.7 ω 2 0 T / [ 1 + ( ω T ) 2 ] .
E ( z , t ) = ( 1 / a ) ( t - z / c ) sin [ ω ( t - z / c ) ] for 0 < ( t - z ) / c < a = sin [ ω ( t - z / c ) ] for a < ( t - z / c ) < ( τ - a ) = { 1 - ( 1 / a ) [ t - z / c - ( τ - a ) ] } sin [ ω ( t - z / c ) ] for ( τ - a ) < ( t - z / c ) < τ = 0 for τ < ( t - z / c ) < L ,
E ( z , t ) = n = 1 ( A n 2 + B n 2 ) 1 / 2 sin [ λ n ( t - z / c ) + ϕ n ] ,
a L A n = [ sin ( ω + λ n ) a + sin ( ω + λ n ) ( τ - a ) - sin ( ω + λ n ) τ ] / ( ω + λ n ) 2 + [ sin ( ω - λ n ) a + sin ( ω - λ n ) ( τ - a ) - sin ( ω - λ n ) τ ] / ( ω - λ n ) 2 , a L B n = [ 1 - cos ( ω + λ n ) a + cos ( ω + λ n ) τ - cos ( ω + λ n ) ( τ - a ) ] / ( ω + λ n ) 2 + [ - 1 + cos ( ω - λ n ) a - cos ( ω - λ n ) τ + cos ( ω - λ n ) ( τ - a ) ] / ( ω - λ n ) 2 ,
limit a 0 A n = 2 ω ( 1 - cos λ n τ ) L ( ω 2 - λ n 2 ) , limit a 0 B n = - 2 ω sin λ n τ / L ( ω 2 - λ n 2 ) .
E ( z , t ) = n = 1 C n sin [ λ n ( t - z / c ) + ϕ n ] ,
C n = - 4 ω sin ( λ n τ / 2 ) / L ( ω 2 - λ n 2 ) , ϕ n = - λ n τ / 2.
× E = - B / t , × H = J + D / t , · D = 0 , · H = 0.
J ( r , t ) = 0 W J ( u ) E ( r , t - u ) d u ,
D ( r , t ) = 0 W D ( u ) E ( r , t - u ) d u ,
2 ξ s / z 2 + κ 2 ξ s = 0 ,
κ 2 = λ 2 μ 0 - i λ μ 0 σ .
= Re ( W ˜ D ) + Im ( W ˜ J ) / ω
σ = Re ( W ˜ J ) - ω Im ( W ˜ D ) ,
β = λ ( μ 0 / 2 ) 1 / 2 { 1 + [ 1 + ( σ / λ ) 2 ] 1 / 2 } 1 / 2
α = λ ( μ 0 / 2 ) 1 / 2 { - 1 + [ 1 + ( σ / λ ) 2 ] 1 / 2 } 1 / 2 .
ξ trans ( z , t ) = 2 E i ( β I - i α I ) ( β I - i α I ) + ( β II - i α II ) × exp [ - α II z + i ( λ t - β II z ) ] ,
E trans = 2 E i exp ( - α II z ) sin ( λ t - β II z + θ ) [ ( 1 + β II c / λ ) 2 + ( α II c / λ ) 2 ] 1 / 2 ,
tan θ = α II / [ ( λ / c ) + β II ] .
E trans ( z , t ) = 2 n = 1 ( A n 2 + B n 2 ) 1 / 2 exp ( - α II , n z ) sin ( λ t - β II , n z + ϕ n + θ n ) [ ( 1 + β II , n c / λ ) 2 + ( α II , n c / λ ) 2 ] 1 / 2 .
R 513 ( z , t ) = 2 n = 513 C n exp ( - α II , n z ) sin ( λ t - β II , n z + ϕ n + θ n ) [ ( 1 + β II , n c / λ ) 2 + ( α II , n c / λ ) 2 ] 1 / 2 .
R 513 ( z , t ) 2 n = 513 4 ω L ( λ n 2 - ω 2 ) < 8 ω L ( λ 513 2 - ω 2 ) + 8 ω L 513 d n ( λ n 2 - ω 2 ) .
R 513 ( z , t ) < 8 ω L ( λ 513 2 - ω 2 ) - 2 π ln λ 513 - ω λ 513 + ω .

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