Abstract

In Part I of this two-part investigation [J. Opt. Soc. Am. A 22, 1200 (2005)], we presented a theory for phase-space propagation of time-harmonic electromagnetic fields in an anisotropic medium characterized by a generic wave-number profile. In this Part II, these investigations are extended to transient fields, setting a general analytical framework for local analysis and modeling of radiation from time-dependent extended-source distributions. In this formulation the field is expressed as a superposition of pulsed-beam propagators that emanate from all space–time points in the source domain and in all directions. Using time-dependent quadratic-Lorentzian windows, we represent the field by a phase-space spectral distribution in which the propagating elements are pulsed beams, which are formulated by a transient plane-wave spectrum over the extended-source plane. By applying saddle-point asymptotics, we extract the beam phenomenology in the anisotropic environment resulting from short-pulsed processing. Finally, the general results are applied to the special case of uniaxial crystal and compared with a reference solution.

© 2005 Optical Society of America

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References

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  1. I. Tinkelman, T. Melamed, “Local spectrum analysis of field propagation in anisotropic media. Part I. Time-harmonic fields,” J. Opt. Soc. Am. A, 22, 1200–1207 (2005).
    [CrossRef]
  2. E. Heyman, L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
    [CrossRef]
  3. E. Heyman, L. B. Felsen, “Weakly dispersive spectraltheory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
    [CrossRef]
  4. T. B. Hansen, A. D. Yaghjian, “Planar near-field scanning in the time-domain. Part I: fomulation,” IEEE Trans. Antennas Propag. 42, 1280–1291 (1994).
    [CrossRef]
  5. T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
    [CrossRef]
  6. E. Heyman, T. Melamed, “Space-time representation of ultra windband signals,” in Advances in Imaging and Electron Physics (Academic, San Diego, Calif., 1998), Vol. 103, pp. 3–63.
  7. E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
    [CrossRef]

2005 (1)

1997 (1)

T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
[CrossRef]

1994 (2)

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

T. B. Hansen, A. D. Yaghjian, “Planar near-field scanning in the time-domain. Part I: fomulation,” IEEE Trans. Antennas Propag. 42, 1280–1291 (1994).
[CrossRef]

1987 (2)

E. Heyman, L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman, L. B. Felsen, “Weakly dispersive spectraltheory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

Felsen, L. B.

E. Heyman, L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman, L. B. Felsen, “Weakly dispersive spectraltheory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

Hansen, T. B.

T. B. Hansen, A. D. Yaghjian, “Planar near-field scanning in the time-domain. Part I: fomulation,” IEEE Trans. Antennas Propag. 42, 1280–1291 (1994).
[CrossRef]

Heyman, E.

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

E. Heyman, L. B. Felsen, “Weakly dispersive spectraltheory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

E. Heyman, L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman, T. Melamed, “Space-time representation of ultra windband signals,” in Advances in Imaging and Electron Physics (Academic, San Diego, Calif., 1998), Vol. 103, pp. 3–63.

Melamed, T.

I. Tinkelman, T. Melamed, “Local spectrum analysis of field propagation in anisotropic media. Part I. Time-harmonic fields,” J. Opt. Soc. Am. A, 22, 1200–1207 (2005).
[CrossRef]

T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
[CrossRef]

E. Heyman, T. Melamed, “Space-time representation of ultra windband signals,” in Advances in Imaging and Electron Physics (Academic, San Diego, Calif., 1998), Vol. 103, pp. 3–63.

Tinkelman, I.

Yaghjian, A. D.

T. B. Hansen, A. D. Yaghjian, “Planar near-field scanning in the time-domain. Part I: fomulation,” IEEE Trans. Antennas Propag. 42, 1280–1291 (1994).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

T. B. Hansen, A. D. Yaghjian, “Planar near-field scanning in the time-domain. Part I: fomulation,” IEEE Trans. Antennas Propag. 42, 1280–1291 (1994).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

E. Heyman, L. B. Felsen, “Weakly dispersive spectral theory of transients (STT), part I: formulation and interpretation,” IEEE Trans. Antennas Propag. 35, 80–86 (1987).
[CrossRef]

E. Heyman, L. B. Felsen, “Weakly dispersive spectraltheory of transients (STT), part II: evaluation of the spectral integral,” IEEE Trans. Antennas Propag. 35, 574–580 (1987).
[CrossRef]

E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

J. Electromagn. Waves Appl. (1)

T. Melamed, “Phase-space beam summation: a local spectrum analysis for time-dependent radiation,” J. Electromagn. Waves Appl. 11, 739–773 (1997).
[CrossRef]

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

Other (1)

E. Heyman, T. Melamed, “Space-time representation of ultra windband signals,” in Advances in Imaging and Electron Physics (Academic, San Diego, Calif., 1998), Vol. 103, pp. 3–63.

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Figures (6)

Fig. 1
Fig. 1

Transient plane-wave spectrum. (a) The transient plane-wave distribution u ̃ + o ( ξ , τ ) is obtained by Radon transform (8) of the initial field distribution u + o ( x , t ) over surfaces of linear delay t c 1 ξ x = const . (b) Transient plane-wave propagating in the direction κ ̂ ( ξ ) of (10).

Fig. 2
Fig. 2

Local (phase-space) spectrum and local beam coordinates. The PS spectral distribution is obtained by windowing of u o ( x , t ) in the three-dimensional ( x , t ) domain with a time-dependent window w ( x , t ) , which is shifted to the space– time point ( x ¯ , t ¯ ) and is tilted by a linear delay of ξ ¯ ( x x ¯ ) . Thus the PS transform extracts local directional properties of u o ( x , t ) and by that matches a single PB propagator emanating from the window center at ( x ¯ , t ¯ ) along κ ¯ ̂ .

Fig. 3
Fig. 3

PB on-axis temporal distribution. The solid curve plots the asymptotic on-axis field for z b = 0.005 as a function of time, and circles plot the reference solution.

Fig. 4
Fig. 4

Snapshots of (a) the asymptotic and (b) the reference field in the ( x b 1 , z b ) plane. (c) Relative error in decibels for points where the reference field is more than 30 db from its on-axis peak.

Fig. 5
Fig. 5

Snapshots of PB propagator on-axis PB distributions as a function of z b for (a) the near ( t = 0.05 ) and (b) the far ( t = 1 ) field. The solid curves depict the asymptotic propagator, and circles represent the reference solution.

Fig. 6
Fig. 6

Contour plots of 1 , 3 , and 6 db from peak level of the off-axis PB-propagator distribution in the ( x b 1 , z b ) cross-sectional plane, for both the asymptotic (solid curves) and the reference field (circles). The dashed curve represents the radius of curvature.

Equations (55)

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ζ = ζ ( ξ ) , ξ = ( ξ 1 , ξ 2 ) .
u ̂ ( r , ω ) = d t u ( r , t ) exp ( i ω t ) ,
u ( r , t ) = 1 2 π d ω u ̂ ( r , ω ) exp ( i ω t ) d ω .
u + ( r , t ) = 1 π 0 d ω u ̂ ( r , ω ) exp ( i ω t ) , Im t 0 ,
u ( r , t ) = Re u + ( r , t ) , t real .
u ̃ + o ( ξ , τ ) = 1 π 0 d ω u ̃ ̂ o ( ξ , ω ) exp ( i ω τ ) ,
u ̃ + o ( ξ , τ ) = d 2 x 1 π 0 d ω u ̂ o ( x , ω ) exp [ i ω ( τ + c 1 ξ x ) ] ,
u ̃ + o ( ξ , τ ) = d 2 x u + o ( x , τ + c 1 ξ x ) .
u + ( r , t ) = ( 2 π c ) 2 d 2 ξ t 2 u ̃ + o { ξ , t c 1 [ ξ x + ζ ( ξ ) z ] } ,
κ ̂ ( ξ ) = [ ξ , ζ ( ξ ) ] ( ξ ξ + ζ 2 ) 1 2 .
U o ( Y ¯ ) = 1 2 π d ω U ̂ o ( X ¯ , ω ) exp ( i ω t ¯ ) ,
U o ( Y ¯ ) = d 2 x d t u o ( x , t ) W ( x , t ; Y ¯ ) ,
W ( x , t ; Y ¯ ) = w [ x x ¯ , t t ¯ c 1 ξ ¯ ( x x ¯ ) ]
U o ( Y ¯ ) = ( 2 π c ) 2 d 2 ξ d τ u ̃ o ( ξ , τ ) τ 2 w ̃ ( ξ ξ ¯ , τ t ¯ + c 1 ξ x ¯ ) ,
u ( x , t ) = ( 2 π c ) 2 d 5 Y ¯ U ( Y ¯ ) W N ( x , t ; Y ¯ ) ,
W N ( x , t ; Y ¯ ) = N 1 ( t ) W ( x , t ; Y ¯ ) ,
N 1 ( t ) = 1 2 π d ω ( i ω ) 2 N ̂ 2 ( ω ) exp ( i ω t ) ,
N ( t ) = t 2 d 2 x w ( x , t ) w ( x , t ) ,
u ( r , t ) = ( 2 π c ) 2 d 5 Y ¯ U o ( Y ¯ ) B ( r , t ; Y ¯ ) ,
B + ( r , t ; Y ¯ ) = ( 2 π c ) 2 d 2 ξ t 2 W ̃ + N [ ξ , t c 1 ( ξ x + ζ ( ξ ) z ) ; Y ¯ ] ,
W ̃ + N ( ξ , τ ; Y ¯ ) = 1 π 0 d ω [ i ω N ̂ ( ω ) ] 2 W ̃ ̂ ( ξ ; X ¯ , ω ) exp [ i ω ( τ t ¯ ) ] ,
W ̃ + N ( ξ , τ ; Y ¯ ) = N 1 ( t ) w ̃ + ( ξ ξ ¯ , τ t ¯ + c 1 ξ x ¯ ) ,
B + ( r , t ; Y ¯ ) = 1 π 0 d ω [ i ω N ̂ ( ω ) ] 2 B ̂ ( r ; X , ω ) exp [ i ω ( t t ¯ ) ] .
T ω max 1 ,
w ̂ ( x ) = ( i ω ) 2 exp ( 1 2 ω T + i 2 k o x Γ x T ) ,
N ̂ 2 ( ω ) = c π ω 3 exp ( ω T ) Γ i .
N ̂ 2 ( ω ) = c π ω 3 Γ i , ω ω max .
w ( x , t ) = Re w + ( x , t ) = Re δ + ( t i 2 T c 1 1 2 x Γ x T ) ,
ζ ( ξ ) = ( 1 ξ 2 ) 1 2 , ξ 2 ξ ξ , Im ζ 0 .
B + ( r , t ; Y ¯ ) = [ det Γ ( z b ) det Γ ( 0 ) ] 1 2 i Γ i π c 1 π 0 d ω ( i ω ) exp { i ω [ t t ¯ i 2 T c 1 S ( r ) ] } ,
S ( r b ) = z b + 1 2 x b Γ ( z b ) x b T .
B + ( r , t ; Y ¯ ) = i Γ i π c [ det Γ ( z b ) det Γ ( 0 ) ] 1 2 δ + [ t t ¯ τ ( r b ) ] ,
τ ( r b ) = i 2 T + c 1 S ( r b )
Γ 1 , 2 ( z b ) = 1 R 1 , 2 ( z b ) + i I 1 , 2 ( z b ) ,
R 1 , 2 = ( z b Z 1 , 2 ) + F 1 , 2 2 ( z b Z 1 , 2 ) ,
I 1 , 2 ( z b ) = F 1 , 2 [ 1 + ( z b Z 1 , 2 ) 2 F 1 , 2 2 ] ,
τ ( r b ) = t p ( r b ) + ( i 2 ) T p ( r b ) ,
t p ( r b ) = c 1 [ z b + x b 1 2 ( 2 R 1 ) + x b 2 2 ( 2 R 2 ) ] ,
T p ( r b ) = T + c 1 ( x b 1 2 I 1 + x b 2 2 I 2 ) .
D 1 , 2 ( z b ) = 2 [ ( 2 1 ) c T I 1 , 2 ( z b ) ] 1 2 .
( x x ¯ ) R ¯ = ξ ¯ , t ¯ = t c 1 R ¯ ,
B ̂ ( r ; X ¯ ) = ( i ω ) 2 [ det Γ ( z b ) det Γ ( 0 ) ] 1 2 exp { i ω [ i 2 ω T c 1 S ( r b ) ] } ,
S ( r b ) = κ ¯ ̂ iso [ r ( x ¯ , 0 ) ] + 1 2 x b Γ ( z b ) x b T ,
B + ( r , t ; Y ¯ ) = 1 π 0 d ω ω Γ i c π [ det Γ ( z b ) det Γ ( 0 ) ] 1 2 exp { i ω [ t t ¯ + i 2 ω T c 1 S ( r b ) ] } .
B ( r , t ; Y ¯ ) = Re i Γ i π c [ det Γ ( z b ) det Γ ( 0 ) ] 1 2 δ + [ t t ¯ τ ( r b ) ] ,
τ ( r b ) = i 2 T + c 1 S ( r b )
Re i δ + [ t t ¯ t p i 2 T p ] = 1 π ( t t ¯ t p ) 2 ( T p 2 ) 2 [ ( t t ¯ t p ) 2 + ( T p 2 ) 2 ] 2 ,
t p ( r ) = c 1 ( κ ¯ ̂ iso r + x b 1 2 2 R 1 + x b 2 2 2 R 2 ) ,
T p ( r ) = T + c 1 ( x b 1 2 I 1 + x b 2 2 I 2 ) ,
I 1 , 2 ( z b ) = a 1 , 2 F 1 , 2 [ 1 + ( z b Z 1 , 2 ) 2 F 1 , 2 2 ] ,
D 1 , 2 ( z b ) = 2 [ ( 2 1 ) c T I 1 , 2 ( z b ) ] 1 2 ,
( x x ¯ ) R ¯ = cos ϑ ¯ 1 , 2 , t ¯ = t c 1 [ x x ¯ 2 + ( z ) 2 ] 1 2 ,
W ̃ ̂ ( ξ ; X ¯ ) = i ω 2 π c Γ exp ( i ω { i T 2 + c 1 [ 1 2 ( ξ ξ ¯ ) Γ 1 ( ξ ξ ¯ ) T + ξ x ¯ ] } ) ,
W ̃ + N ( ξ , τ ; Y ¯ ) = 2 i Γ i Γ δ + { τ t ¯ + i T 2 + c 1 [ 1 2 ( ξ ξ ¯ ) Γ 1 ( ξ ξ ¯ ) T + ξ x ¯ ] } .
ζ ( ξ ) = [ 1 ( ξ 2 2 + ξ 1 2 ) ϵ z ] 1 2 , c = c 0 ϵ .

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