Abstract

We introduce a body-of-revolution finite-difference time-domain simulation capability that can be applied to rotationally symmetric linear-optics problems. This simulator allows us to reduce a computationally intractable, three-dimensional problem to a numerically solvable two-dimensional one. It is used to model the propagation of a pulsed Gaussian beam through a thin dielectric lens and the focusing of the resulting pulsed beam. Analytic results for such a lens-focused, pulsed Gaussian beam are also derived. It is shown that, for the same input energy, one can design ultrawide-bandwidth driving signals to achieve a significantly larger intensity enhancement than is possible with equivalent many-cycle, monochromatic signals. Several specially engineered (designer) pulses are introduced that illustrate how one can achieve these intensity enhancements. The simulation results confirm that intensity enhancements can be realized with properly designed ultrawide-bandwidth pulses.

© 1994 Optical Society of America

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References

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  1. R. W. Ziolkowski, J. B. Judkins, “Full-wave vector Maxwell equation modelling of the self-focusing of ultrashort optical pulses in a nonlinear Kerr medium exhibiting a finite response time,” J. Opt. Soc. Am. B 10, 186–198 (1993).
    [CrossRef]
  2. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag., AP-14302–307 (1966).
  3. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
  4. A. Taflove, K. R. Umashankar, “The finite-difference time-domain method for numerical modelling of electromagnetics wave interactions with arbitrary structures,” in Finite Element and Finite Difference Methods in Electromagnetic Scattering, M. A. Morgan, ed., Vol. 2 of Progress in Electromagnetic Research (Elsevier, New York, 1990).
  5. D. E. Merewether, R. Fisher, “Finite difference solution of Maxwell’s equation for EMP applications,” Rep. EMA-79–R-4 (Defense Nuclear Agency, Washington, D.C., 1980).
  6. A. C. Cangellaris, R. Lee, “On the accuracy of numerical wave simulations based on finite methods,” J. Electromagn. Waves Appl. 6, 1635–1653 (1992).
    [CrossRef]
  7. R. W. Ziolkowski, J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
    [CrossRef]
  8. D. B. Davidson, “A parallel processing tutorial,” IEEE Antennas Propag. Mag. 32(4), 6–19 (1990).
    [CrossRef]
  9. D. B. Davidson, “Parallel processing revisited: a second tutorial,” IEEE Antennas Propag. Mag. 34(10), 9–21 (1992).
    [CrossRef]
  10. R. F. Harrington, Field Computation by Moment Methods, 2nd ed. (Krieger, Malabar, Fla., 1982).
  11. A. Taflove, K. R. Umashankar, “Advanced numerical modeling of microwave penetration and coupling for complex structures—final report,” final Rep. UCRL-15960, Contract 6599805 (Lawrence Livermore National Laboratory, Livermore, Calif., 1987).
  12. J. Bailey, “Implementing fine-grained scientific algorithms on the Connection Machine supercomputer,” Thinking Machines Tech. Rep. Ser. TR90-1 (Thinking Machines Corporation, Cambridge, Mass., 1990).
  13. R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” IEEE Trans. Antennas Propag. 40, 888–905 (1992).
    [CrossRef]
  14. R. W. Ziolkowski, I. M. Besieris, A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
    [CrossRef]
  15. G. C. Sherman, “Short pulses in the focal region,” J. Opt. Soc. Am. A 6, 1382–1387 (1989).
    [CrossRef]
  16. Z. Bor, Z. L. Horváth, “Distortion of femtosecond laser pulse in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
    [CrossRef]
  17. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986).
  18. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).
  19. I. S. Gradshteyn, I. M. Ryzhik, eds., Tables of Integrals, Series and Products (Academic, New York, 1965).

1993 (2)

1992 (5)

D. B. Davidson, “Parallel processing revisited: a second tutorial,” IEEE Antennas Propag. Mag. 34(10), 9–21 (1992).
[CrossRef]

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” IEEE Trans. Antennas Propag. 40, 888–905 (1992).
[CrossRef]

A. C. Cangellaris, R. Lee, “On the accuracy of numerical wave simulations based on finite methods,” J. Electromagn. Waves Appl. 6, 1635–1653 (1992).
[CrossRef]

R. W. Ziolkowski, J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
[CrossRef]

Z. Bor, Z. L. Horváth, “Distortion of femtosecond laser pulse in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

1990 (1)

D. B. Davidson, “A parallel processing tutorial,” IEEE Antennas Propag. Mag. 32(4), 6–19 (1990).
[CrossRef]

1989 (1)

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag., AP-14302–307 (1966).

Bailey, J.

J. Bailey, “Implementing fine-grained scientific algorithms on the Connection Machine supercomputer,” Thinking Machines Tech. Rep. Ser. TR90-1 (Thinking Machines Corporation, Cambridge, Mass., 1990).

Besieris, I. M.

Bor, Z.

Z. Bor, Z. L. Horváth, “Distortion of femtosecond laser pulse in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Cangellaris, A. C.

A. C. Cangellaris, R. Lee, “On the accuracy of numerical wave simulations based on finite methods,” J. Electromagn. Waves Appl. 6, 1635–1653 (1992).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

Davidson, D. B.

D. B. Davidson, “Parallel processing revisited: a second tutorial,” IEEE Antennas Propag. Mag. 34(10), 9–21 (1992).
[CrossRef]

D. B. Davidson, “A parallel processing tutorial,” IEEE Antennas Propag. Mag. 32(4), 6–19 (1990).
[CrossRef]

Fisher, R.

D. E. Merewether, R. Fisher, “Finite difference solution of Maxwell’s equation for EMP applications,” Rep. EMA-79–R-4 (Defense Nuclear Agency, Washington, D.C., 1980).

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods, 2nd ed. (Krieger, Malabar, Fla., 1982).

Horváth, Z. L.

Z. Bor, Z. L. Horváth, “Distortion of femtosecond laser pulse in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Judkins, J. B.

Lee, R.

A. C. Cangellaris, R. Lee, “On the accuracy of numerical wave simulations based on finite methods,” J. Electromagn. Waves Appl. 6, 1635–1653 (1992).
[CrossRef]

Merewether, D. E.

D. E. Merewether, R. Fisher, “Finite difference solution of Maxwell’s equation for EMP applications,” Rep. EMA-79–R-4 (Defense Nuclear Agency, Washington, D.C., 1980).

Shaarawi, A. M.

Sherman, G. C.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986).

Taflove, A.

A. Taflove, K. R. Umashankar, “The finite-difference time-domain method for numerical modelling of electromagnetics wave interactions with arbitrary structures,” in Finite Element and Finite Difference Methods in Electromagnetic Scattering, M. A. Morgan, ed., Vol. 2 of Progress in Electromagnetic Research (Elsevier, New York, 1990).

A. Taflove, K. R. Umashankar, “Advanced numerical modeling of microwave penetration and coupling for complex structures—final report,” final Rep. UCRL-15960, Contract 6599805 (Lawrence Livermore National Laboratory, Livermore, Calif., 1987).

Umashankar, K. R.

A. Taflove, K. R. Umashankar, “Advanced numerical modeling of microwave penetration and coupling for complex structures—final report,” final Rep. UCRL-15960, Contract 6599805 (Lawrence Livermore National Laboratory, Livermore, Calif., 1987).

A. Taflove, K. R. Umashankar, “The finite-difference time-domain method for numerical modelling of electromagnetics wave interactions with arbitrary structures,” in Finite Element and Finite Difference Methods in Electromagnetic Scattering, M. A. Morgan, ed., Vol. 2 of Progress in Electromagnetic Research (Elsevier, New York, 1990).

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag., AP-14302–307 (1966).

Ziolkowski, R. W.

IEEE Antennas Propag. Mag. (2)

D. B. Davidson, “A parallel processing tutorial,” IEEE Antennas Propag. Mag. 32(4), 6–19 (1990).
[CrossRef]

D. B. Davidson, “Parallel processing revisited: a second tutorial,” IEEE Antennas Propag. Mag. 34(10), 9–21 (1992).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag., AP-14302–307 (1966).

R. W. Ziolkowski, “Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays,” IEEE Trans. Antennas Propag. 40, 888–905 (1992).
[CrossRef]

J. Electromagn. Waves Appl. (1)

A. C. Cangellaris, R. Lee, “On the accuracy of numerical wave simulations based on finite methods,” J. Electromagn. Waves Appl. 6, 1635–1653 (1992).
[CrossRef]

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

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

Opt. Commun. (1)

Z. Bor, Z. L. Horváth, “Distortion of femtosecond laser pulse in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Other (9)

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975).

I. S. Gradshteyn, I. M. Ryzhik, eds., Tables of Integrals, Series and Products (Academic, New York, 1965).

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

A. Taflove, K. R. Umashankar, “The finite-difference time-domain method for numerical modelling of electromagnetics wave interactions with arbitrary structures,” in Finite Element and Finite Difference Methods in Electromagnetic Scattering, M. A. Morgan, ed., Vol. 2 of Progress in Electromagnetic Research (Elsevier, New York, 1990).

D. E. Merewether, R. Fisher, “Finite difference solution of Maxwell’s equation for EMP applications,” Rep. EMA-79–R-4 (Defense Nuclear Agency, Washington, D.C., 1980).

R. F. Harrington, Field Computation by Moment Methods, 2nd ed. (Krieger, Malabar, Fla., 1982).

A. Taflove, K. R. Umashankar, “Advanced numerical modeling of microwave penetration and coupling for complex structures—final report,” final Rep. UCRL-15960, Contract 6599805 (Lawrence Livermore National Laboratory, Livermore, Calif., 1987).

J. Bailey, “Implementing fine-grained scientific algorithms on the Connection Machine supercomputer,” Thinking Machines Tech. Rep. Ser. TR90-1 (Thinking Machines Corporation, Cambridge, Mass., 1990).

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Figures (15)

Fig. 1
Fig. 1

The lens used in the numerical simulations was double parabolic and thin. For all the cases the lens is located at z = 3.0 μm, it is 2.25 μm thick at its base, and its index of refraction is n = 2. For the F# = 1.0 cases the focal length f = 18.0 μm and the diameter D = 18.0 μm. For the F# = 0.707 case f = 9.0 μm and D = 12.732 μm.

Fig. 2
Fig. 2

Contour plot of the Eρ-field distribution at the time step n = 150 for the 1-cycle, F# = 0.707 case. The pulsed beam is mainly interacting with the lens at this time. Some energy has already been reflected from the front face of the lens. The relative sizes of the beam waist and the lens radius are apparent.

Fig. 3
Fig. 3

Contour plot of the Eρ-field distribution at the time step n = 300 for the 1-cycle, F# = 0.707 case. The pulsed beam has passed through the lens and is now focusing. The change in curvature of the wave fronts caused by the interaction of the pulsed beam with the lens is apparent.

Fig. 4
Fig. 4

Contour plot of the Eρ -field distribution at the time step n = 450 for the 1-cycle, F# = 0.707 case. The pulsed beam is at the focus. The decrease in the beam waist and the time derivative of the field are immediately apparent.

Fig. 5
Fig. 5

Contour plot of the Eρ -field distribution at the time step n = 900 for the 1-cycle, F# = 0.707 case. The pulsed beam is well beyond the focus. The increase in the beam waist, the change in the wave-front curvature, and the leading wave front caused by the edge of the lens are readily identifiable.

Fig. 6
Fig. 6

Signal along the propagation axis at the focus should be nearly the time derivative of the original source signal. The FDTD numerically generated signal at the focus is compared with the time derivative of the source signal. Solid curve, time derivative of the source signal; dashed curve, FDTD-computed pulse.

Fig. 7
Fig. 7

FFT of the FDTD numerically generated signal at the focus compared with the FFT of the time derivative of the source signal. Solid curves, time derivative of the source signal; dashed curve, FDTD-computed pulse.

Fig. 8
Fig. 8

Intensity in the simulation space contour plotted as a function of the distance along the axis of propagation and of the time. This is a spectrogramlike plot of the field distribution during the simulation. The location of the focus and the evolution of the derivative behavior of the field are clearly apparent with this visualization approach.

Fig. 9
Fig. 9

10-cycle, 1-cycle, and UWB1 IE pulses plotted as functions of time. The 1-cycle pulse has a time record length identical to one period of the 10-cycle pulse. The positive-to-negative switch time is slightly less than one half of one period of the 10-cycle case. Solid curve, UWB1 pulse; dashed curve, 10-cycle pulse; dotted curve, 1-cycle pulse.

Fig. 10
Fig. 10

FFT’s of the 10-cycle, the 1-cycle, and the UWB1 IE pulses plotted as functions of the frequency. Solid curve, UWB1 pulse; dashed curve, 10-cycle pulse; dotted curve, 1-cycle pulse.

Fig. 11
Fig. 11

IE values of the 10-cycle, the 1-cycle, and the UWB1 IE pulsed beams focused by the thin dielectric lens plotted as functions of the distance along the propagation axis. Solid curve, UWB1; dotted–dashed curve, 1-cycle pulse; dotted curve, 10-cycle pulse.

Fig. 12
Fig. 12

Waists of the 10-cycle, the 1-cycle, and the UWB1 IE pulsed beams focused by the thin dielectric lens plotted as functions of the distance along the propagation axis. Solid curve, UWB1; dotted–dashed curve, 1-cycle pulse; dotted curve, 10-cycle pulse.

Fig. 13
Fig. 13

Contour plot of the Eρ -field distribution at the time step n = 750 in the focal region for the 1-cycle, F# = 1.0 case. The pulsed beam is at the focus.

Fig. 14
Fig. 14

Contour plot of the Eρ -field distribution at the time step n = 3732 in the focal region for the UWB1 designer pulse, F# = 1.0 case. The pulsed beam is at the focus.

Fig. 15
Fig. 15

Mesh showing interleaved field components.

Tables (2)

Tables Icon

Table 1 Comparison of the Focal Lengths, Waist Radii, and Energy-Enhancement Factors Predicted Analytically with the Corresponding Values Obtained from the Energies Computed Numerically by Use of the BOR-FDTD Codea

Tables Icon

Table 2 Comparison of the Focal Lengths, Waist Radii, and Intensity-Enhancement Factors Predicted Analytically with the Corresponding Values Obtained from the Intensities Computed Numerically by Use of the BOR–FDTD Codea

Equations (118)

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E inc = E ( t k ̂ R c ) ( ŷ × k ̂ ) .
k ̂ = sin ( θ ) x ̂ cos ( θ ) ,
k ̂ = sin ( θ ) cos ( ϕ ) ρ ̂ sin ( θ ) sin ( ϕ ) ϕ ̂ cos ( θ ) ,
ŷ × k ̂ = cos ( θ ) cos ( ϕ ) ρ ̂ cos ( θ ) sin ( ϕ ) ϕ ̂ + sin ( θ ) ,
R = ρ ρ ̂ + z ,
k ̂ R = ρ sin ( θ ) cos ( ϕ ) z cos ( θ ) .
E ρ ( r , z , ϕ , t ) = E ρ 0 ( r , z , t ) + k = 1 E ρ k ( r , z , t ) cos ( k ϕ ) ,
E ρ 0 ( r , z , t ) = 1 2 π 0 2 π E ρ ( r , z , ϕ , t ) d ϕ ,
E ρ k ( r , z , t ) = 1 π 0 2 π E ρ ( r , z , ϕ , t ) cos ( k ϕ ) d ϕ .
E ϕ ( r , z , ϕ , t ) = E ϕ 0 ( r , z , t ) + k = 1 E ϕ k ( r , z , t ) sin ( k ϕ ) ,
E ϕ k ( r , z , t ) = 1 π 0 2 π E ϕ ( r , z , ϕ , t ) sin ( k ϕ ) d ϕ .
E inc = f ( x , y ) E [ t ( z / c ) ] x ̂ ,
E inc = f ( ρ ) E [ t ( z / c ) ] [ cos ( ϕ ) ] ρ ̂ sin ( ϕ ) ϕ ̂ ] .
Δ t 1 υ max { 1 Δ ρ 2 [ ( m + 1 ) 2 + 2.8 ] 4 + 1 Δ z 2 } 1 / 2 .
H ϕ k n + 1 ( i , j ) = H ϕ k n ( i , j ) Δ t μ × { E ρ k n ( i , j + 1 ) [ E ρ k n ( i , j ) + E p k n ( i , j ) z 0 ( j + 1 ) z 0 ( j ) } + Δ t μ [ E z k n ( i + 1 , j ) E z k n ( i , j ) ρ 0 ( i + 1 ) ρ 0 ( i ) ] ,
B E ρ k n + 1 ( i , j ) = A E ρ k n ( i , j ) + k ρ ( i ) H z k n + 1 ( i , j + 1 ) × { [ H ϕ k n + 1 ( i , j + 1 ) H ϕ k n + 1 ( i , j + 1 ) H ϕ k n + 1 ( i , j ) ] z ( j + 1 ) z ( j ) } J ρ k ,
E ρ k n ( i , j ) = { f ( t ) exp ( ρ 2 / ω 0 2 ) if j = j sact / tot 0 otherwise ,
H ϕ k n + 1 ( i , j ) = { f ( t t r ) exp ( ρ 2 / ω 0 2 ) if j = j sact / tot 0 otherwise .
t r = Δ z 2 c ( 1 ξ / 2 ) .
[ z 1 c t ] ϕ ( z , t ) = 0 .
z ϕ ( z , t ) | z = 0 = 1 c t ϕ ( z , t ) | z = 0 .
ϕ 1 n ϕ 0 n = Δ z c Δ t ( ϕ 0 n + 1 ϕ 0 n ) ,
ϕ 0 n + 1 = ϕ 0 n ( 1 c Δ z Δ z ) + c Δ t Δ z ϕ 1 n .
ϕ k max + 1 n + 1 = ϕ k max + 1 n ( 1 c Δ t Δ z ) + c Δ t Δ z ϕ k max n ,
G ( ρ , ϕ , z , t ) = E 0 exp ( ρ 2 / w 0 2 ) F ( t ) ,
F ( ω ) = F ( t ) exp ( + i ω t ) d t ,
F ( t ) = 1 2 π F ( ω ) exp ( i ω t ) d ω .
G ( ρ , ϕ , z , ω ) = E 0 exp ( ρ 2 / w 0 2 ) F ( ω ) .
ω rad 2 def = A d S d t | G ( r , t ) / t | 2 A d S d t | G ( r , t ) | 2 = A d S d ω ω 2 | G ( r , ω ) | 2 A d S d ω | G ( r , ω ) | 2 .
ω rad 2 = d t | F ( t ) / t | 2 d t | F ( t ) | 2 = d ω ω 2 | F ( ω ) | 2 d ω | F ( ω ) | 2 .
g ( r , t ) A d S [ 2 G ( x , y , z = 0 , t ) ( c t ) ] t = t R / c 1 4 π R ,
in = d t | G ( ρ = 0 , ϕ , z = 0 , t ) | 2 ,
rad ( z ) = d t | g ( ρ = 0 , ϕ , z , t ) | 2 .
L G = π w 0 2 / λ rad
rad G ( z ) in G ( L G z ) 2 .
w enrg ( z ) ~ θ enrg G z ~ λ rad z π w 0 ,
θ enrg G ~ λ rad π w 0 .
θ int G θ enrg G ,
T lens ( ρ , ϕ , z , ω ) = exp ( i ω ρ 2 2 c f ) 2 .
g ( r , t ) 1 2 π d ω A d S T lens ( r , ω ) ( 2 i ω c ) × G ( r , ω ) exp [ i ω ( t R / c ) ] 4 π R .
g ( r , t ) 1 2 π d ω F ( ω ) 0 2 π d ϕ × 0 a d ρ ρ exp [ i ( ω / c ) ( ρ 2 / 2 f ) ] × 1 4 π R { E 0 ( 2 i ω c ) exp ( ρ 2 / w 0 2 ) × exp [ i ω ( t R / c ) ] } = 2 E 0 2 π c t d ω F ( ω ) × 0 2 π d ϕ 0 a d ρ ρ exp [ i ( ω / c ) ( ρ 2 / 2 f ) ] { exp ( ρ 2 / w 0 2 ) exp [ i ω ( t R / c ) ] 4 π R } .
R = [ ρ 2 + ρ 2 2 ρ ρ cos ( ϕ ϕ ) + z 2 ] 1 / 2 ~ z + ρ 2 + ρ 2 2 z ρ ρ z cos ( ϕ ϕ ) ,
g ( r , t ) E 0 2 π c z t d ω F ( ω ) exp [ i ω ( t z / c ) ] × exp [ + i ( ω / c ) ( ρ 2 / 2 z ) ] × 0 a d ρ ρ exp [ i ( ω / c ) ( ρ 2 / 2 f ) ] × exp [ + i ( ω / c ) ( ρ 2 / 2 z ) ] exp ( ρ 2 / w 0 2 ) × 1 2 π 0 2 π d ϕ exp [ + i ( ω / c ) ( ρ ρ / z ) cos ( ϕ ϕ ) ] = E 0 2 π c z t d ω F ( ω ) exp [ i ω ( t z / c ) ] × exp [ + i ( ω / c ) ( ρ 2 / 2 z ) ] × 0 a d ρ ρ exp ( ρ 2 / w 0 2 ) × exp [ i ( ω c ) ( 1 z 1 f ) ( ρ 2 2 ) ] J 0 ( ω c ρ z ρ ) .
g ( ρ = 0 , ϕ , z , t ) E 0 2 π c z t d ω F ( ω ) × exp [ i ω ( t z / c ) ] × 0 a d ρ ρ exp [ ( Λ / ω 0 2 ) ρ 2 ] = E 0 w 0 2 2 c z t d ω 2 π F ( ω ) × exp [ i ω ( t z / c ) ] × 1 exp [ ( a / w 0 ) 2 Λ ] Λ = E 0 L G z 1 ω rad t d ω 2 π F ( ω ) × exp [ i ω ( t z / c ) ] × 1 exp [ ( a / w 0 ) 2 Λ ] Λ ,
Λ = 1 i ω w 0 2 2 c ( 1 z 1 f ) = 1 i ω ω rad L G ( 1 z 1 f )
g ( ρ = 0 , ϕ , z , t ) ~ E 0 L G z 1 ω rad t d ω 2 π F ( ω ) × exp [ i ω ( t z / c ) ] × { 1 exp [ ( a / w 0 ) 2 ] Λ } ~ E 0 L G z 1 ω rad { 1 exp [ ( a / w 0 ) 2 ] } × t F ( t z / c ) + E 0 L G L R z ( 1 z 1 f ) exp [ ( a / w 0 ) 2 ] × 1 ω rad 2 2 t 2 F ( t z / c ) ,
g ( ρ = 0 , ϕ , z = f , t ) ~ E 0 L G f 1 ω rad × t d ω 2 π F ( ω ) exp [ i ω ( t f / c ) ] × { 1 exp [ ( a / w 0 ) 2 ] } = E 0 L G f 1 ω rad { 1 exp [ ( a / w 0 ) 2 ] } × t F ( t f / c ) .
A enrg def = rad ( f ) input = { 1 exp [ ( a / w 0 ) 2 ] } 2 ( L G f ) 2 .
A int def = max t | g ( ρ = 0 , ϕ , z = f , t ) | 2 max t | G ( ρ = 0 , ϕ , z = 0 , t ) | 2 ~ { 1 exp [ ( a / w 0 ) 2 ] } 2 ( L G f ) 2 × [ max t | F ( t f / c ) / t | 2 ω rad 2 = ϒ int × A enrg ,
ϒ int × [ max t | F ( t f / c ) / t | 2 ω rad 2 [ max t | F ( t ) / t | 2 ] ω rad 2 .
A enrg cw = A enrg UWB ~ ( L G f ) 2 ,
A int UWB F A int cw = [ max t | F UWB ( t ) / t | 2 ] ω rad 2 = ϒ int UWB .
0 d x exp ( α x 2 ) J 0 ( β x ) = exp ( β 2 / 4 α ) 2 α ( Re α > 0 ) ,
0 a d ρ ρ exp [ ( ρ / w 0 ) 2 ] J 0 ( ω c ρ f ρ ) = 1 2 exp { 1 4 [ ( ω c z ) 2 ] ρ 2 } 0 d ρ ρ exp [ ( ρ / w 0 ) 2 ] J 0 ( ω c ρ f ρ ) .
g ( ρ , ϕ , z = f , t ) E 0 L G f 1 ω rad t d ω 2 π F ( ω ) × exp [ i ω ( t f / c ρ 2 / 2 c f ) ] × exp [ ω 2 ( w 0 ρ / 2 c f ) 2 ] = E 0 L G f 1 ω rad t F ( t f / c ρ 2 / 2 c f ) * t ( π t exp { ( w rad t ) 2 [ ( L G f ) ( ρ w 0 ) ] 2 } ) ,
d x exp ( i α x ) exp ( β 2 x 2 ) = π α exp ( α 2 / 4 β 2 ) .
g ( ρ , ϕ , z = f , t ) E 0 L G f 1 ω rad t F ( t f / c ρ 2 / 2 c f ) ,
w ( f ) ( f L G ) w 0 = θ enrg G f ,
F PL ( t ) = { t / T 1 for 0 t T 1 1 2 ( t T 1 ) ( T 2 T 1 ) for T 1 t T 2 ( t T 3 ) / ( T 3 T 2 ) for T 2 t T 3 .
t F PL ( t ) = { 1 / T 1 for 0 t T 1 2 / ( T 2 T 1 ) for T 1 t T 2 1 / ( T 3 T 2 ) for T 2 t T 3 .
d t | F PL | 2 = T 3 2 = ( m + 1 2 ) δ ,
d t | t F PL | 2 = 1 T 1 + 4 T 2 T 1 + 1 T 3 T 2 = ( 4 + 2 m ) 1 δ ,
max t | t F PL | 2 = 4 δ 2 ,
ω rad , PL 2 = 4 m 1 δ 2 = 4 ϒ int PL 1 δ 2 ,
ϒ int PL = m .
F TR ( t ) = { t / T 1 for 0 t T 1 1.0 for T 1 t T 2 1 2 ( t t 2 ) / ( T 3 T 2 ) for T 2 t T 3 1.0 for T 3 t T 4 ( t T 5 ) / ( T 5 T 4 ) for T 4 t T 5 .
t F TR ( t ) = { t / T 1 for 0 t T 1 0.0 for T 1 t T 2 2 / ( T 2 T 1 ) for T 2 t T 3 0.0 for T 3 t T 4 1 / ( T 5 T 4 ) for T 4 t T 5 .
d t | F TR | 2 = 1 2 ( 2 n + 1 ) δ + 2 m δ = ( 2 m + 2 n + 1 2 ) δ ,
d t | t F TR | 2 = 1 T 1 + 4 T 2 T 1 + 1 T 3 T 2 = ( 4 + 2 2 ) 1 δ ,
max t | t F TR | 2 = 4 δ 2 ,
ω rad , TR 2 = 4 ϒ int TR 1 δ 2 ,
ϒ int TR = 4.0 2 m + ( 2 n + 1 ) / 2 4 + 2 / n = 2 m + ( 2 n + 1 ) / 2 1 + 1 / 2 n .
F IE ( t ) = { ( 1 x 2 ) 4 , x = 1.0 + t / T 1 for 0 t T 1 1 + 2 ( 1 x 2 ) 4 , x = ( t T 2 ) / ( T 2 T 1 ) for T 1 t T 2 ( 1 x 2 ) 4 , x = ( t T 2 ) / ( T 3 T 2 ) for T 2 t T 3 .
t F IE ( t ) = { [ 8 / T 1 ] x ( 1 x 2 ) 3 , x = 1.0 + t / T 1 for 0 t T 1 [ 16 / ( T 2 T 1 ) ] x ( 1 x 2 ) 3 , x = ( t T 2 ) / ( T 2 T 1 ) for T 1 t T 2 [ 8 / ( T 3 T 2 ) ] x ( 1 x 2 ) 3 , x = ( t T 2 ) / ( T 3 T 2 ) for T 2 t T 3 .
F CW ( t ) = [ 1 x ( t ) 2 ] 4 × sin [ m ( 2 π t / T ) ] ( windowed cw ) ,
F SC ( t ) = ( 16 / T ) x ( t ) [ 1 x ( t ) 2 ] 3 ( single cycle ) ,
F UWB 3 ( t ) = ( 16 / T ) x ( t ) [ 1 x ( t ) 2 ] 3 × { 1 + [ β x ( t ) ] 2 } α × ( 1 + 0.25 α β 2 [ 1 x ( t ) 2 ] { 1 + [ β x ( t ) ] 2 } 1 ) ( UWB 3 ) ,
x ( t ) = 1 2 ( t / T ) .
f = a 2 2 d ( n 1 ) .
μ H ρ k t = ( k ρ E z k E ϕ k z ) ,
μ H ϕ k t = ( E ρ k z E z k ρ ) ,
μ H z k t = 1 ρ [ ( ρ E ϕ k ) ρ + k E ρ k ] ,
E ρ k t + σ E ρ k = ( k ρ H z k H ϕ k z ) J ρ k ,
E ϕ k t + σ E ϕ k = ( H ρ k z H z k ρ ) J ϕ k ,
E z k t + σ E z k = 1 ρ [ ( ρ H ϕ k ) ρ k H ρ k ] J z k .
H ρ k n ( i , j ) = H ρ k [ ρ 0 ( i ) , z ( j ) , t H ( n ) ] ,
H ϕ k n ( i , j ) = H ϕ k [ ρ ( i ) , z ( j ) , t H ( n ) ] ,
H z k n ( i , j ) = H z k [ ρ ( i ) , z 0 ( j ) , t H ( n ) ] ,
E ρ k n ( i , j ) = E ρ k [ ρ ( i ) , z 0 ( j ) , t E ( n ) ] ,
E ϕ k n ( i , j ) = E ϕ k [ ρ 0 ( i ) , z 0 ( j ) , t E ( n ) ] ,
E z k n ( i , j ) = E z k [ ρ 0 ( i ) , z ( j ) , t E ( n ) ] ,
C = C 0 + k = 1 k = k max C k cos ( k ϕ ) .
D = k = 1 k = k max D k sin ( k ϕ ) .
ρ 0 ( i ) = ( i 1 ) Δ ρ + ρ c ,
ρ ( i ) = ( i ½ ) Δ ρ + ρ c ,
z 0 ( j ) = ( j 1 ) Δ z ,
z ( j ) = ( j ½ ) Δ z ,
t H ( n ) = ( n 1 ) Δ t ,
t E ( n ) = ( n ½ ) Δ t ,
H ρ k n + 1 ( i , j ) = H ρ k n ( i , j ) + k Δ t μ ρ 0 ( i ) E z k n ( i , j ) + Δ t μ [ E ϕ k n ( i , j + 1 ) E ϕ k n ( i , j ) z 0 ( j + 1 ) z 0 ( j ) ] ,
H ϕ k n + 1 ( i , j ) = H ϕ k n ( i , j ) Δ t μ [ E ρ k n ( i , j + 1 ) E ρ k n ( i , j ) z 0 ( j + 1 ) z 0 ( j ) ] + Δ t μ [ E z k n ( i + 1 , j ) E z k n ( i , j ) ρ 0 ( i + 1 ) ρ 0 ( i ) ] ,
H z k n + 1 ( i , j ) = H z k n ( i , j ) Δ t μ { ρ 0 ( i + 1 ) E ϕ k n ( i + 1 , j ) ρ 0 ( i ) E ϕ k n ( i , j ) ρ ( i ) [ ρ 0 ( i + 1 ) ρ 0 ( i ) ] } k Δ t μ ρ ( i ) E ρ k n ( i , j ) ,
B E ρ k n + 1 ( i , j + 1 ) = A E ρ k n ( i , j + 1 ) + k ρ ( i ) H z k n + 1 ( i , j + 1 ) [ H ϕ k n + 1 ( i , j + 1 ) H ϕ k n + 1 ( i , j ) z ( j + 1 ) z ( j ) ] J ρ k ,
B E ϕ k n + 1 ( i + 1 , j + 1 ) = A E ϕ k n ( i + 1 , j + 1 ) + [ H ρ k n + 1 ( i + 1 , j + 1 ) H ρ k n + 1 ( i + 1 , j ) z ( j + 1 ) z ( j ) ] [ H z k n + 1 ( i + 1 , j + 1 ) H z k n + 1 ( i , j + 1 ) ρ ( i + 1 ) ρ ( i ) ] J ϕ k ,
B E z k n + 1 ( i + 1 , j ) = A E z k n ( i + 1 , j ) + { ρ ( i + 1 ) H ϕ k n + 1 ( i + 1 , j ) ρ ( i ) H ϕ k n + 1 ( i , j ) ρ 0 ( i + 1 ) [ ρ ( i + 1 ) ρ ( i ) ] } k ρ 0 ( i + 1 ) H ρ k n + 1 ( i + 1 , j ) J z k ,
A = ( Δ t σ 2 ) ,
B = ( Δ t + σ 2 ) ,
J ρ k = J ρ k ( t H n + 1 ) ,
J ϕ k = J ϕ k ( t H n + 1 ) ,
J z k = J z k ( t H n + 1 ) ,
E z k t + σ E z k = 1 ρ [ ( ρ H ϕ k ) ρ k H ρ k ] J z k .
E z k t E z k n + 1 ( i , j ) E z k n ( i , j ) Δ t ,
( ρ H ϕ k ) ρ ρ ( i + 1 ) H ϕ k n + 1 ( i + 1 , j ) ρ ( i ) H ϕ k n + 1 ( i , j ) ρ 0 ( i + 1 ) [ ρ ( i + 1 ) ρ ( i ) ] .
S × H = C H dl = S ( E t + J + σ E ) dS ,
0 2 π H ϕ 0 ρ δ ϕ = 0 2 π 0 ρ ( E z 0 t + J z 0 + σ E z 0 ) ρ δ ϕ δ ρ ,
B E z 0 n + 1 ( 1 , j ) = A E z 0 n ( 1 , j ) + 2 H ϕ 0 n + 1 ( 1 , j ) ρ ( 1 ) J z 0 ( 1 , j ) j [ 1 , j max ] .
E z k ( 1 , j ) = 0 k [ 1 , k max ] and j [ 1 , j m a x ] ,
E ϕ k ( 1 , j ) = 0 k [ 1 , k max ] and j [ 1 , j m a x ] .

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