- Email: [email protected]
Theory for the propagation of short electromagnetic pulses
a __
__ !!B
ELSEVIER
15 June 1997 OPTICS COMMUNICATIONS Optics Communications
139 (1997) I-6
Theory for the propagation of short electromagnetic pulses Nicholas George
I,
CenrerfiJr Electronic Imaging S_vsremu.The lnstitufe of Optics. Received
Stojan Radic Utzicvrsig
of Rochester. Rochester, NY 14627. USA
18 November 1996; accepted 8 January
1997
Abstract A theory is presented for the free-space propagation of short pulses of any duration, including sub-cycle or unipolar pulses. From Maxwell’s equations, we provide an exact solution of the vector potential and the scalar potential which result for an electric dipole created stepwise at t = 0 with a corresponding Dirac-delta in source current density. This idealized unipulse provides valuable insight as to the separation of 1/r propagating terms from those of higher order arising from the scalar potential. Secondly, an exact solution is presented for the radiation field in temporal form which arises from an assertion of an input aperture distribution. Since the temporal assertion may lead to propagation terms of l/r dependence and others of higher order, we illustrate how to make an informed choice. A finite-difference-time-domain computer study of four selected sub-cycle pulses is used to illustrate the method and to verify our prediction that free-space propagation is a linear filter of temporal frequency varying as [iZrrv/(rc)] expt - i?rrvr/c) with a null at zero frequency.
1. Introduction With the advent of ultra-wideband pulse radars, considerable attention has been directed to various aspects of the generation, propagation. and reception of this electromagnetic radiation. Since the free-space propagation of an ultra-wideband (UWB) pulse is within the domain of Maxwell’s equations, it is important to present a rigorous framework for the solution of the many important problems arising. This is handy as a benchmark for comparison of results using the important computer method finite-difference time-domain (FDTD) and the equally important methods of Fourier optics. In this material we present two different idealized problems dealing with the propagation of ultrashort unipolar pulses. First, as an idealized but realizable source, we consider the creation of an electric dipole, using the Heaviside step function U(r). by a unipolar Dirac-delta function impulse of current density J located at the origin. This source distribution is helpful to develop understanding since it provides insight into the propagation of the I/r fields and in contrast to the static field as well. Also,
I
E-mail: [email protected].
interestingly, the static field is readily separated since the electric dipole remains “on” as t --) cc. Next, we describe the calculation for radiation from an aperture using an “assumed” distribution of electric field. This type of source problem is often termed a secondary source. We show that some care needs to be exercised in this assertion of a specific temporal behavior. The field can have propagating and higher order components. In any event we are able to write a theoretical form for the solution of the radiation from an open aperture. This will be recognized as a fruitful avenue for further research. It is clear from this example that one can rewrite the vector diffraction integral equivalents of Maxwell’s equations in a time-dependent fashion. For brevity in our presentation, we started from the well-known Rayleigh-SommerfeldSmythe form. However, for theoretical purposes it would be more satisfying to start with a time-dependent form of Maxwell’s equations. Appealing theoretical avenues are the single and double-sided Laplace transforms [ 1,2]. Related early literature on an electric dipole stepped on at t = 0 is contained in Sommerfeld’s lectures [3] and in an article by B. van der Pol 141. Equations for pulsed antennas and for the transient response of an infinitesimal dipole of moment p(t) are given in a monograph by C.H. Papas [5]. Two other references have been found in which a Dirac-
0030-4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved p/l SOO30-40 I8(97)00038-2
2
N. George, S. Radic/
Optics Communications
delta impulse is used to turn on the electric dipole [6,7]. In these latter two, the current is no longer unipolar. They use a Coulomb-gauge, and so the relevance to the UWB propagation is not as direct.
139 f 1997) 1-6
A, the well-known exact differential radiation field are given by [8] 1 V’A - pe?
equations
for the
= -pJ(x,Y,z,r),
(5)
2
2. Propagation from an impulse in current
v*-
Consider an electric dipole of charges fq, separated by a tiny distance za, located at the origin, and aligned along the Z-axis. This dipole is created in a stepwise fashion at r = 0. Hence, the charge density p(x, y,z,r) is given by
in rationalized mks units with the speed of light, L’= ( /Je)- “2. In free space the general solution of Eqs. (5) and (6) can be expressed in the form of time-retarded potentials, as follows:
P( x,Y,Z,r)
A( x,y,~,r)
=
qoqx)S(
zo/q-
Y>[ q z -
6( z +
z,/2)]U(r),
PC::
=-
(1) in which the Heaviside step-function
U(t)=
u(t)
L
-
J(X1.Y ,,z,,r-r,/L’) dx, dy, dz,,
4%-///
rl
T(x,,,z,t) =-
dr
= qo6( x)S( y)rect(
1
P(x,,Y,*z,,r-r,/u) dx, dy, di,,
4%-e Jil
rI (8)
‘,=[(X-xx,)~+(Y-y,)~+(z-Z,)~]“2.
dp( X,Y,ZJ)
Hence, with Eqs. (l)-(3) applied to the dipole shown in Fig. 1, we can readily deduce that the corresponding source current is given by J( x-,~,z,t)
(6)
(7)
The relation between the current-density vector J and the charge density is given by the equation of continuity which expresses the conservation of charge, viz.,
V.J(x,y,z,t)=
-p(x,.v,i,t)/E,
is defined as
t>o, t < 0.
;,
CL
=
(9)
Note that the integration is over the region of space which contains sources. From these potentials, one recovers the magnetic induction B and the electric field vector E using the expressions B=p,H=VXA
z/+)8(r)
e,,
(4)
in which eL is the unit vector along the z-axis and s(t) is Dirac’s delta function, i.e., an idealized unipolar pulse which results from the differentiation of U(f) in Eq. (1). The function rect( z/z,) is unity when 1Z/Q/ < l/2 and zero outside of this interval. From the time-dependent form of Maxwell’s equations using the Lorentz gauge in specifying the vector potential
(‘0)
and E=
-z-w.
(11)
The localized form of the sources greatly simplifies the calculations. Eq. (4) is substituted in Eq. (7) and directly integrated. Similarly Eq. (1) into (8) is computed with more algebra. For the Dirac-unipolar impulse in J, we find the following vector and scalar potentials, viz., eLA;(x,y,z,~)
= F
6(t-r/u)e,,
(12)
z
~(x,y,z,t)=~
=o
+qo -+
-
JZ t,
-t Fig. I. An idealized unipolar impulse in current density creates a step in charge, see Eqs. (2) and (4).
cl020 [
U(t - r/u) r2
+
s(t-r/u) ru
l2
r’
(13)
From this solution it is evident that at a distance r, the vector potential is directly proportional and aligned with the source current density at a retarded time. Likewise the scalar potential has two identifiable terms; the first clearly gives the usual “static” potential due to a simple electric dipole as t + m. From Eqs. (12) and (13) using Eqs. (10) and (1 l), one can compute all of the field components at
N. George. S. Rndic / Optics Communications 139 (19971 l-6
an arbitrary position f x,v,z). In the interest of brevity, we will calculate only the spherical component of the electric field, &, where H is the spherical polar coordinate of an (r, 0, I#J) system. This transverse component of electric field serves as a good indicator of the entire propagation field. For the Dirac-impulse source current, the transverse component of the electric field EH is given by
+ q,,:, he
U(t - r/r) [
ri
+
S(r - r/lx)
rZ L:
Note that the derivative of the delta function, represented by the following limit form:
sin@.
(14)
I
= lim Z u-0 7r ($ + o2)? ’
r,~lr-r’/=\/(x-xx’)‘+(y-y’)Z+z”.
(17)
exp( - i kr, l/r, is the free-space Green’s function, n is a unit normal pointing in the fz direction, and &r’) is the exact electric field in the aperture. The integration extends only over the aperture because the screen is a perfect conductor. In the signal representation of the electric vector f?(r’), we have used the “tilde” to specify that we are dealing with the temporal Fourier transform of the time varying electric field. Moreover, we emphasize that the time-vary ing function has been written in an analytic signal form. Hence, in definition of the signal notation, one can write
S’(t), can be i(r;v)
crt S’(r)
where
cu>o.
(‘5)
= jX E,(x,~,z;r)e-‘2a”‘dr, -*
and the Fourier inversion: E,(x,y,z;t)
Clearly by Eq. (15). the derivative of the Dirac-delta is seen to have a large positive lobe at negative argument followed by a large negative lobe for positive argument. Now in Eq. (14) only the term falling off with distance as l/r represents propagation (in the radar sense). At large distances the scalar potential (last 2 members) does not contribute a meaningful signal. Most interesting in Eq. (14) is the absence of a unipolar-component (dc) in the propagating term with the 1/r dependence. This is evident from the bipolar nature of the S’(t) in Eq. (1.5). Hence from Eqs. (41, (I 2) and (14), we have established that the radiation field due to a Dirac-delta-unipulse in current density leads to a “following” unipulse in the vector potential at retarded time but to a bipolar impulse in the electric field.
(18)
= jX E(r;Y)e+“““dV. -r
(‘9)
The subscript ‘a’ denotes an analytic signal form. Note as well that the harmonic time dependent analysis using exp(iot) would yield a formula that is notationally as given by Eq. (16) with 27r~ = o. It is useful to write out explicitly the expressions for all three components of the electric field transforms: exp( - ikr,) #Q x,y,z)
= ;
j j ~.~(X’,Y’,O) A
TI
(20)
X(ik+-l-)(;i-)dr’d;‘,
k,(x,.v,z)
= ;
j jAE,(+‘,LO)
exp( - i kr, ) r,
x(ik+-!-)(t)di.dr.,
3. Radiation from apertures (time-dependent) For an infinitesimally thin perfectly conducting sheet in the plane at z = 0 with an open aperture letting radiation from sources (; < 0) propagate into the right-half-space, one can calculate the radiation for z > 0 in terms of the tangential electric field in the aperture. The coordinate axes here should not be confused with the choice used in the earlier sections. Here, we are considering the z-axis as the axis of propagation, as is customary in the RayleighSommerfeld-Smythe formulation. The electric field E(r) in the right half-space, i.e., z 2 0, is then uniquely specified by the tangential component of the exact electric field in the aperture 19, IO]:
(‘6)
E;(x,y,z>
= &
/jJ
(Y)
E,(x',
+ (+)
X
(21) G’,V’~O)
y’,O)]
exp( - ikr, ) rl
(22)
Note too that k = 27rv/u and for an UWB pulse this variable changes dramatically. Hence, the output ,!?,( x, y, z 1 is governed by !o%,
N. George, S. Radic/
Fig. 2. Configuration
for FDTD
Oprics Communications
calculation with slit width )I’=
5.55A and distance P = 14.44h.
Eq. (19). For purposes of illustration, it is adequate to invert Eqs. (16) and (21). Hence, radiation can be calculated from the following aperture forms: E,( x,.v,z;r) R
x E,( x’,y’,O;t - r-,/u) dx’ dy’, YI (23)
+
E,,( x’,y’,O;t 27rrF
1
r,/u>
dx’dy’, (24)
it is instructive to compare Eqs. (23) and (24). As a typical problem, one would like to assert a time-dependent form at the aperture and then calculate the fields propagating to an arbitrary point (x, y,z), as in Fig. 2. From Eq. (23) we have the well-known operation of retarded time r + I I-,/u. By the expansion of Eq. (24) one observes that the scalar component of tangential electric field can be separated into a term containing a/&[ E?,] that propagates with l/r range dependence and a second term containing [E?,] that propagates as l/r’. In summary for an arbitrary aperture distribution E?.,(x,y,O;t), we see that the radiation field will have a bipolar form as evidenced by the time derivative a[ E,.,]/& in Eq. (24).
4. Finite-difference time-domain calculation of nearfield evolution for incident unipolar signal Starting from Yee’s original idea [ 111, finite-difference time-domain (FDTD) methods have played an important role in the establishment of a number of rigorous solutions of Maxwell’s equations. During the 1960s and 1970s FDTD methods have been successfully used for complex EM scattering problems that include radar cross sections,
139 (IY!?7) 1-6
human tissue imaging, and high-speed circuit design [ 12141. In the 1980s and more recently, an increasingly important role has emerged for the FDTD approach in ultrafast optoelectronics and photonics [ 15 191. A number of important advances that include generalizations encompassing dispersive and nonlinear materials [ 15,161 have allowed modeling of soliton propagation, ultrafast switching, and dispersive periodic structures [ 18,191. Combined with current developments in ultrafast optics that lead to sub-cycle pulse generation and breakdown of any approximation (slow-envelope, small perturbation strength) the generalization of FDTD methods will serve as a preferred, and often the only rigorous tool at our disposal. Dispersive or nonlinear properties of the medium are readily incorporated into the method as we previously described [ 191. The physical limitation to such a rigorous approach is three-fold: (a) computing resource (speed), (b) available memory storage, and (c) boundary condition implementation. Even as we are writing these lines, an improvement in condition (a) is being made by a new generation of fast parallel processing architectures. Condition (b) currently limits our calculation to 50 X 50 wavelength size in 2-D and 15 X 15 X 15 wavelength size in 3-D for a very fast (non-buffered) computation. Finally, condition (c) has been a subject of substantial research in the last two decades [20-221, yielding some exceptional boundary algorithms. The role of FDTD boundary condition is to limit computation size without introducing any artificial (numerical) diffraction at the boundary of the modeling site. We are currently using Engquist-Majda [22] condition which introduces an error of - 10-j. In this research on the diffraction and propagation of subwavelength pulses, Eqs. (23) and (24) permit one to select ad-hoc pulse envelopes to use as secondary sources. However, when we use an assumed value for the tangential field, it often will contain terms that propagate as (l/r) and higher order terms dropping off as l/r2. We believe that FDTD methods are ideally suited to test this notion, since they provide an exact numerical solution. Terms arising, in effect, from a scalar potential will drop extremely fast. Typically by Eq. (201, one can estimate that in the near zone the ratio of the propagation term to the near-field term will be increased on the order of h. To test the validity of this notion, we need to examine the evolution of an assumed subwavelength pulse in the near field. We expect the form to change rapidly at first, thereby placing in evidence a bipolar remainder term which is essentially entirely a propagation term falling off more gradually as (l/r). For the FDTD calculation, consider an aperture in a conducting plane at z = 0 with the radiation incident from the left, shown in Fig. 2. For an incident plane wave (A = 0.9 mm) diffracted by a large aperture (5.55h wide), we calculate the resulting field after propagation to a point P, 14.44A away. At P, this field history is stored and used to calculate spectra, as shown in Figs. 3a-3d. As illus-
5
N. George, S. Radic / Optics Communications 139 (1997) 1-6
temporal Fourier transform, e.g., in Eq. (18), then it is appropriate to consider that h (x, y; v> is a “transfer-function” insofar as its temporal origin, i.e., in the v variable. This interpretation lends clarity to the spectral curves in Fig. 3, since we see that the l/r term has a transfer function proportional to the term i2nu exp( - i2nv r/c). Insofar as the temporal frequency dependence, this zero in the transfer function explains the intuitive filtering-out of the “static or dc component” hypothesized in the initial aperture distribution. Again the same observation can be drawn for the idealized unipolar pulse in Eq. (4). The radiation term given by the retarded vector potential [A] in Eq. (7) has the &function form. Hence in computing the corresponding electric field by taking a time derivative leads to the s’(t) in Eqs. (14) and (15). Finally, considering this S’(r) as an impulse response, we take the temporal Fourier transform to find the transfer function. Since ,7 6’(r) = i2av we find a consistent explanation.
trated in our earlier theory, the temporal dependence consists of a propagating (“dynamic”) portion and a “static” one. Our assertion is that FDTD calculation of the field at R, > 14A is an excellent demonstration of “static” field filtering. Figs. 2a-2c show inputs (dashed) which are synthesized with a strong unipolar component. The solid curve shows considerable modification due to the filtering of the near-field components. Alternatively, Fig. 2d shows a less drastic modification due to the initial choice of a bipolar form; i.e., one complete cycle of a sine wave.
5. Impulse response and transfer function for free-space From Eqs. (20) and (2 1) in Fourier optics, it is customto note the perfect convolution of the input, I?,(x, y,O;v), with the normal derivative of Sommerfeld’s choice for the Green’s function. This function is termed the impulse response h(x - .r’,v - $;v) and is given in Ref. [23] by ary
exp( -ikr) h( x,v;v)
=
*nr
2
Acknowledgements
i2rv
,(,+$
P)
The authors are pleased to acknowledge the helpful comments by B.D. Guenther and the support of the Army Research Office and the National Science Foundation.
in which r = (.I-’ + x’ + ;‘)“‘. However, if we consider the temporal dependence of the input E,,(x, v.O;r) and the
Ki
a 440
46.0
46.0
50.0
52.0
I
& 440
46.0
.o
-1
50.0
520
0.0
0.5
1
I
I
1.0 1
1~~,,~~
b
48.0
-0.5
-1
.o
-0.5
a.0
0.5
1.0
/
.
-0.5
0.0
0.5
1.0
0.0
0.5
1.0
C
44.0
46.0
46.0
50.0
52.0
-1
11
d Ti
44.0
46.0
48.0
50.0
52.0
.o
-1
.o
-0.5
Fig. 3. Finite-difference time-domain solutions for the pulse shapes (dashed) in the aperture shown in Fig. 2. Corresponding spectra are also shown (dashed in aperture and solid at P).
after propagation
(solid) to P at ; = 14.44h,
6
N. George, S. Radic/Optics
References [I] H.S. Carslaw and J.C. Jaeger, Operational Methods in Applied Mathematics, 2nd Ed. (Oxford University Press, London, 1947). [2] B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Transform, 2nd Ed. (Cambridge University Press, London, 1955). [3] A. Sommerfeld, Vorlesungen ilber theoretische Physik, Band III, Electrodynamik (Wiesbaden, 1948) p. 150. [4] B. van der Pal, IRE Trans. Antennas Propag. AP-4 (1956) 288. [5] C.H. Papas, Yerevan Lectrures on Electromagnetic Theory (California Institute of Technology Press, Pasadena, 1972) Ch. 9. [6] O.L. Brill and B. Goodman, Amer. J. Physics 35 (1967) 832. [7] J.D. Jackson. Classical Electrodynamics, 2nd Ed. (Wiley, New York, 1975) prob. 6.19, p. 267. [S] W.R. Smythe, Static and Dynamic Electricity, 3rd Ed., revised (Summa-Hemisphere, New York, 1989) Ch. XII.
Communications [9] [lo] [I 11 [12] [13] [14] [15] [16] [ 171 [18] [19] [20] [21] 1221 [23]
139 11997) 1-6
W.R. Smythe, Phys. Rev. 72 (1947) 1066. R.E. English and N. George, Appl. Optics 26 (1987) 2362. K.S. Yee, IEEE Trans. Antennas Propag. AP-14 (1966) 302. K. Umashaukar and A. Taflove, IEEE Trans. Electromag. Compat. EMC-24 (1982) 397. KS. Kunz and K.M. Lee, IEEE Trans. Electromag. Compat. EMC-20 (1978) 328. A. Taflove, Computational Electrodynamics: The FDTD Method (Artech House, 1995). R.M. Joseph, S.G. Hagness and A. Taflove, Optics Lett. 18 (1991) 1412. P.M. Goorjian, A. Taflove, R.M. Joseph and S. Hagness, IEEE J. Quantum Electron. 28 (1992) 2416. A. Taflove, Wave Motion (1988) p. 547. R.M. Joseph and P.M. Goorjian. Optics Lett. 48 (1993) 491. S. Radic and N. George, Optics Lett. 19 (1994) 1064. J.G. Blaschak, J. Comput. Phys. 77 (1988) 109. J.P. Berenger, J. Comput. Phys. 114 (1994) 185. B. Engquist and A. Majda, Math. Comput. 31 (1977) 629. N. George, Optics Comm. 133 (1997) 22.
__ !!B
ELSEVIER
15 June 1997 OPTICS COMMUNICATIONS Optics Communications
139 (1997) I-6
Theory for the propagation of short electromagnetic pulses Nicholas George
I,
CenrerfiJr Electronic Imaging S_vsremu.The lnstitufe of Optics. Received
Stojan Radic Utzicvrsig
of Rochester. Rochester, NY 14627. USA
18 November 1996; accepted 8 January
1997
Abstract A theory is presented for the free-space propagation of short pulses of any duration, including sub-cycle or unipolar pulses. From Maxwell’s equations, we provide an exact solution of the vector potential and the scalar potential which result for an electric dipole created stepwise at t = 0 with a corresponding Dirac-delta in source current density. This idealized unipulse provides valuable insight as to the separation of 1/r propagating terms from those of higher order arising from the scalar potential. Secondly, an exact solution is presented for the radiation field in temporal form which arises from an assertion of an input aperture distribution. Since the temporal assertion may lead to propagation terms of l/r dependence and others of higher order, we illustrate how to make an informed choice. A finite-difference-time-domain computer study of four selected sub-cycle pulses is used to illustrate the method and to verify our prediction that free-space propagation is a linear filter of temporal frequency varying as [iZrrv/(rc)] expt - i?rrvr/c) with a null at zero frequency.
1. Introduction With the advent of ultra-wideband pulse radars, considerable attention has been directed to various aspects of the generation, propagation. and reception of this electromagnetic radiation. Since the free-space propagation of an ultra-wideband (UWB) pulse is within the domain of Maxwell’s equations, it is important to present a rigorous framework for the solution of the many important problems arising. This is handy as a benchmark for comparison of results using the important computer method finite-difference time-domain (FDTD) and the equally important methods of Fourier optics. In this material we present two different idealized problems dealing with the propagation of ultrashort unipolar pulses. First, as an idealized but realizable source, we consider the creation of an electric dipole, using the Heaviside step function U(r). by a unipolar Dirac-delta function impulse of current density J located at the origin. This source distribution is helpful to develop understanding since it provides insight into the propagation of the I/r fields and in contrast to the static field as well. Also,
I
E-mail: [email protected].
interestingly, the static field is readily separated since the electric dipole remains “on” as t --) cc. Next, we describe the calculation for radiation from an aperture using an “assumed” distribution of electric field. This type of source problem is often termed a secondary source. We show that some care needs to be exercised in this assertion of a specific temporal behavior. The field can have propagating and higher order components. In any event we are able to write a theoretical form for the solution of the radiation from an open aperture. This will be recognized as a fruitful avenue for further research. It is clear from this example that one can rewrite the vector diffraction integral equivalents of Maxwell’s equations in a time-dependent fashion. For brevity in our presentation, we started from the well-known Rayleigh-SommerfeldSmythe form. However, for theoretical purposes it would be more satisfying to start with a time-dependent form of Maxwell’s equations. Appealing theoretical avenues are the single and double-sided Laplace transforms [ 1,2]. Related early literature on an electric dipole stepped on at t = 0 is contained in Sommerfeld’s lectures [3] and in an article by B. van der Pol 141. Equations for pulsed antennas and for the transient response of an infinitesimal dipole of moment p(t) are given in a monograph by C.H. Papas [5]. Two other references have been found in which a Dirac-
0030-4018/97/$17.00 Copyright 0 1997 Elsevier Science B.V. All rights reserved p/l SOO30-40 I8(97)00038-2
2
N. George, S. Radic/
Optics Communications
delta impulse is used to turn on the electric dipole [6,7]. In these latter two, the current is no longer unipolar. They use a Coulomb-gauge, and so the relevance to the UWB propagation is not as direct.
139 f 1997) 1-6
A, the well-known exact differential radiation field are given by [8] 1 V’A - pe?
equations
for the
= -pJ(x,Y,z,r),
(5)
2
2. Propagation from an impulse in current
v*-
Consider an electric dipole of charges fq, separated by a tiny distance za, located at the origin, and aligned along the Z-axis. This dipole is created in a stepwise fashion at r = 0. Hence, the charge density p(x, y,z,r) is given by
in rationalized mks units with the speed of light, L’= ( /Je)- “2. In free space the general solution of Eqs. (5) and (6) can be expressed in the form of time-retarded potentials, as follows:
P( x,Y,Z,r)
A( x,y,~,r)
=
qoqx)S(
zo/q-
Y>[ q z -
6( z +
z,/2)]U(r),
PC::
=-
(1) in which the Heaviside step-function
U(t)=
u(t)
L
-
J(X1.Y ,,z,,r-r,/L’) dx, dy, dz,,
4%-///
rl
T(x,,,z,t) =-
dr
= qo6( x)S( y)rect(
1
P(x,,Y,*z,,r-r,/u) dx, dy, di,,
4%-e Jil
rI (8)
‘,=[(X-xx,)~+(Y-y,)~+(z-Z,)~]“2.
dp( X,Y,ZJ)
Hence, with Eqs. (l)-(3) applied to the dipole shown in Fig. 1, we can readily deduce that the corresponding source current is given by J( x-,~,z,t)
(6)
(7)
The relation between the current-density vector J and the charge density is given by the equation of continuity which expresses the conservation of charge, viz.,
V.J(x,y,z,t)=
-p(x,.v,i,t)/E,
is defined as
t>o, t < 0.
;,
CL
=
(9)
Note that the integration is over the region of space which contains sources. From these potentials, one recovers the magnetic induction B and the electric field vector E using the expressions B=p,H=VXA
z/+)8(r)
e,,
(4)
in which eL is the unit vector along the z-axis and s(t) is Dirac’s delta function, i.e., an idealized unipolar pulse which results from the differentiation of U(f) in Eq. (1). The function rect( z/z,) is unity when 1Z/Q/ < l/2 and zero outside of this interval. From the time-dependent form of Maxwell’s equations using the Lorentz gauge in specifying the vector potential
(‘0)
and E=
-z-w.
(11)
The localized form of the sources greatly simplifies the calculations. Eq. (4) is substituted in Eq. (7) and directly integrated. Similarly Eq. (1) into (8) is computed with more algebra. For the Dirac-unipolar impulse in J, we find the following vector and scalar potentials, viz., eLA;(x,y,z,~)
= F
6(t-r/u)e,,
(12)
z
~(x,y,z,t)=~
=o
+qo -+
-
JZ t,
-t Fig. I. An idealized unipolar impulse in current density creates a step in charge, see Eqs. (2) and (4).
cl020 [
U(t - r/u) r2
+
s(t-r/u) ru
l2
r’
(13)
From this solution it is evident that at a distance r, the vector potential is directly proportional and aligned with the source current density at a retarded time. Likewise the scalar potential has two identifiable terms; the first clearly gives the usual “static” potential due to a simple electric dipole as t + m. From Eqs. (12) and (13) using Eqs. (10) and (1 l), one can compute all of the field components at
N. George. S. Rndic / Optics Communications 139 (19971 l-6
an arbitrary position f x,v,z). In the interest of brevity, we will calculate only the spherical component of the electric field, &, where H is the spherical polar coordinate of an (r, 0, I#J) system. This transverse component of electric field serves as a good indicator of the entire propagation field. For the Dirac-impulse source current, the transverse component of the electric field EH is given by
+ q,,:, he
U(t - r/r) [
ri
+
S(r - r/lx)
rZ L:
Note that the derivative of the delta function, represented by the following limit form:
sin@.
(14)
I
= lim Z u-0 7r ($ + o2)? ’
r,~lr-r’/=\/(x-xx’)‘+(y-y’)Z+z”.
(17)
exp( - i kr, l/r, is the free-space Green’s function, n is a unit normal pointing in the fz direction, and &r’) is the exact electric field in the aperture. The integration extends only over the aperture because the screen is a perfect conductor. In the signal representation of the electric vector f?(r’), we have used the “tilde” to specify that we are dealing with the temporal Fourier transform of the time varying electric field. Moreover, we emphasize that the time-vary ing function has been written in an analytic signal form. Hence, in definition of the signal notation, one can write
S’(t), can be i(r;v)
crt S’(r)
where
cu>o.
(‘5)
= jX E,(x,~,z;r)e-‘2a”‘dr, -*
and the Fourier inversion: E,(x,y,z;t)
Clearly by Eq. (15). the derivative of the Dirac-delta is seen to have a large positive lobe at negative argument followed by a large negative lobe for positive argument. Now in Eq. (14) only the term falling off with distance as l/r represents propagation (in the radar sense). At large distances the scalar potential (last 2 members) does not contribute a meaningful signal. Most interesting in Eq. (14) is the absence of a unipolar-component (dc) in the propagating term with the 1/r dependence. This is evident from the bipolar nature of the S’(t) in Eq. (1.5). Hence from Eqs. (41, (I 2) and (14), we have established that the radiation field due to a Dirac-delta-unipulse in current density leads to a “following” unipulse in the vector potential at retarded time but to a bipolar impulse in the electric field.
(18)
= jX E(r;Y)e+“““dV. -r
(‘9)
The subscript ‘a’ denotes an analytic signal form. Note as well that the harmonic time dependent analysis using exp(iot) would yield a formula that is notationally as given by Eq. (16) with 27r~ = o. It is useful to write out explicitly the expressions for all three components of the electric field transforms: exp( - ikr,) #Q x,y,z)
= ;
j j ~.~(X’,Y’,O) A
TI
(20)
X(ik+-l-)(;i-)dr’d;‘,
k,(x,.v,z)
= ;
j jAE,(+‘,LO)
exp( - i kr, ) r,
x(ik+-!-)(t)di.dr.,
3. Radiation from apertures (time-dependent) For an infinitesimally thin perfectly conducting sheet in the plane at z = 0 with an open aperture letting radiation from sources (; < 0) propagate into the right-half-space, one can calculate the radiation for z > 0 in terms of the tangential electric field in the aperture. The coordinate axes here should not be confused with the choice used in the earlier sections. Here, we are considering the z-axis as the axis of propagation, as is customary in the RayleighSommerfeld-Smythe formulation. The electric field E(r) in the right half-space, i.e., z 2 0, is then uniquely specified by the tangential component of the exact electric field in the aperture 19, IO]:
(‘6)
E;(x,y,z>
= &
/jJ
(Y)
E,(x',
+ (+)
X
(21) G’,V’~O)
y’,O)]
exp( - ikr, ) rl
(22)
Note too that k = 27rv/u and for an UWB pulse this variable changes dramatically. Hence, the output ,!?,( x, y, z 1 is governed by !o%,
N. George, S. Radic/
Fig. 2. Configuration
for FDTD
Oprics Communications
calculation with slit width )I’=
5.55A and distance P = 14.44h.
Eq. (19). For purposes of illustration, it is adequate to invert Eqs. (16) and (21). Hence, radiation can be calculated from the following aperture forms: E,( x,.v,z;r) R
x E,( x’,y’,O;t - r-,/u) dx’ dy’, YI (23)
+
E,,( x’,y’,O;t 27rrF
1
r,/u>
dx’dy’, (24)
it is instructive to compare Eqs. (23) and (24). As a typical problem, one would like to assert a time-dependent form at the aperture and then calculate the fields propagating to an arbitrary point (x, y,z), as in Fig. 2. From Eq. (23) we have the well-known operation of retarded time r + I I-,/u. By the expansion of Eq. (24) one observes that the scalar component of tangential electric field can be separated into a term containing a/&[ E?,] that propagates with l/r range dependence and a second term containing [E?,] that propagates as l/r’. In summary for an arbitrary aperture distribution E?.,(x,y,O;t), we see that the radiation field will have a bipolar form as evidenced by the time derivative a[ E,.,]/& in Eq. (24).
4. Finite-difference time-domain calculation of nearfield evolution for incident unipolar signal Starting from Yee’s original idea [ 111, finite-difference time-domain (FDTD) methods have played an important role in the establishment of a number of rigorous solutions of Maxwell’s equations. During the 1960s and 1970s FDTD methods have been successfully used for complex EM scattering problems that include radar cross sections,
139 (IY!?7) 1-6
human tissue imaging, and high-speed circuit design [ 12141. In the 1980s and more recently, an increasingly important role has emerged for the FDTD approach in ultrafast optoelectronics and photonics [ 15 191. A number of important advances that include generalizations encompassing dispersive and nonlinear materials [ 15,161 have allowed modeling of soliton propagation, ultrafast switching, and dispersive periodic structures [ 18,191. Combined with current developments in ultrafast optics that lead to sub-cycle pulse generation and breakdown of any approximation (slow-envelope, small perturbation strength) the generalization of FDTD methods will serve as a preferred, and often the only rigorous tool at our disposal. Dispersive or nonlinear properties of the medium are readily incorporated into the method as we previously described [ 191. The physical limitation to such a rigorous approach is three-fold: (a) computing resource (speed), (b) available memory storage, and (c) boundary condition implementation. Even as we are writing these lines, an improvement in condition (a) is being made by a new generation of fast parallel processing architectures. Condition (b) currently limits our calculation to 50 X 50 wavelength size in 2-D and 15 X 15 X 15 wavelength size in 3-D for a very fast (non-buffered) computation. Finally, condition (c) has been a subject of substantial research in the last two decades [20-221, yielding some exceptional boundary algorithms. The role of FDTD boundary condition is to limit computation size without introducing any artificial (numerical) diffraction at the boundary of the modeling site. We are currently using Engquist-Majda [22] condition which introduces an error of - 10-j. In this research on the diffraction and propagation of subwavelength pulses, Eqs. (23) and (24) permit one to select ad-hoc pulse envelopes to use as secondary sources. However, when we use an assumed value for the tangential field, it often will contain terms that propagate as (l/r) and higher order terms dropping off as l/r2. We believe that FDTD methods are ideally suited to test this notion, since they provide an exact numerical solution. Terms arising, in effect, from a scalar potential will drop extremely fast. Typically by Eq. (201, one can estimate that in the near zone the ratio of the propagation term to the near-field term will be increased on the order of h. To test the validity of this notion, we need to examine the evolution of an assumed subwavelength pulse in the near field. We expect the form to change rapidly at first, thereby placing in evidence a bipolar remainder term which is essentially entirely a propagation term falling off more gradually as (l/r). For the FDTD calculation, consider an aperture in a conducting plane at z = 0 with the radiation incident from the left, shown in Fig. 2. For an incident plane wave (A = 0.9 mm) diffracted by a large aperture (5.55h wide), we calculate the resulting field after propagation to a point P, 14.44A away. At P, this field history is stored and used to calculate spectra, as shown in Figs. 3a-3d. As illus-
5
N. George, S. Radic / Optics Communications 139 (1997) 1-6
temporal Fourier transform, e.g., in Eq. (18), then it is appropriate to consider that h (x, y; v> is a “transfer-function” insofar as its temporal origin, i.e., in the v variable. This interpretation lends clarity to the spectral curves in Fig. 3, since we see that the l/r term has a transfer function proportional to the term i2nu exp( - i2nv r/c). Insofar as the temporal frequency dependence, this zero in the transfer function explains the intuitive filtering-out of the “static or dc component” hypothesized in the initial aperture distribution. Again the same observation can be drawn for the idealized unipolar pulse in Eq. (4). The radiation term given by the retarded vector potential [A] in Eq. (7) has the &function form. Hence in computing the corresponding electric field by taking a time derivative leads to the s’(t) in Eqs. (14) and (15). Finally, considering this S’(r) as an impulse response, we take the temporal Fourier transform to find the transfer function. Since ,7 6’(r) = i2av we find a consistent explanation.
trated in our earlier theory, the temporal dependence consists of a propagating (“dynamic”) portion and a “static” one. Our assertion is that FDTD calculation of the field at R, > 14A is an excellent demonstration of “static” field filtering. Figs. 2a-2c show inputs (dashed) which are synthesized with a strong unipolar component. The solid curve shows considerable modification due to the filtering of the near-field components. Alternatively, Fig. 2d shows a less drastic modification due to the initial choice of a bipolar form; i.e., one complete cycle of a sine wave.
5. Impulse response and transfer function for free-space From Eqs. (20) and (2 1) in Fourier optics, it is customto note the perfect convolution of the input, I?,(x, y,O;v), with the normal derivative of Sommerfeld’s choice for the Green’s function. This function is termed the impulse response h(x - .r’,v - $;v) and is given in Ref. [23] by ary
exp( -ikr) h( x,v;v)
=
*nr
2
Acknowledgements
i2rv
,(,+$
P)
The authors are pleased to acknowledge the helpful comments by B.D. Guenther and the support of the Army Research Office and the National Science Foundation.
in which r = (.I-’ + x’ + ;‘)“‘. However, if we consider the temporal dependence of the input E,,(x, v.O;r) and the
Ki
a 440
46.0
46.0
50.0
52.0
I
& 440
46.0
.o
-1
50.0
520
0.0
0.5
1
I
I
1.0 1
1~~,,~~
b
48.0
-0.5
-1
.o
-0.5
a.0
0.5
1.0
/
.
-0.5
0.0
0.5
1.0
0.0
0.5
1.0
C
44.0
46.0
46.0
50.0
52.0
-1
11
d Ti
44.0
46.0
48.0
50.0
52.0
.o
-1
.o
-0.5
Fig. 3. Finite-difference time-domain solutions for the pulse shapes (dashed) in the aperture shown in Fig. 2. Corresponding spectra are also shown (dashed in aperture and solid at P).
after propagation
(solid) to P at ; = 14.44h,
6
N. George, S. Radic/Optics
References [I] H.S. Carslaw and J.C. Jaeger, Operational Methods in Applied Mathematics, 2nd Ed. (Oxford University Press, London, 1947). [2] B. van der Pol and H. Bremmer, Operational Calculus Based on the Two-Sided Laplace Transform, 2nd Ed. (Cambridge University Press, London, 1955). [3] A. Sommerfeld, Vorlesungen ilber theoretische Physik, Band III, Electrodynamik (Wiesbaden, 1948) p. 150. [4] B. van der Pal, IRE Trans. Antennas Propag. AP-4 (1956) 288. [5] C.H. Papas, Yerevan Lectrures on Electromagnetic Theory (California Institute of Technology Press, Pasadena, 1972) Ch. 9. [6] O.L. Brill and B. Goodman, Amer. J. Physics 35 (1967) 832. [7] J.D. Jackson. Classical Electrodynamics, 2nd Ed. (Wiley, New York, 1975) prob. 6.19, p. 267. [S] W.R. Smythe, Static and Dynamic Electricity, 3rd Ed., revised (Summa-Hemisphere, New York, 1989) Ch. XII.
Communications [9] [lo] [I 11 [12] [13] [14] [15] [16] [ 171 [18] [19] [20] [21] 1221 [23]
139 11997) 1-6
W.R. Smythe, Phys. Rev. 72 (1947) 1066. R.E. English and N. George, Appl. Optics 26 (1987) 2362. K.S. Yee, IEEE Trans. Antennas Propag. AP-14 (1966) 302. K. Umashaukar and A. Taflove, IEEE Trans. Electromag. Compat. EMC-24 (1982) 397. KS. Kunz and K.M. Lee, IEEE Trans. Electromag. Compat. EMC-20 (1978) 328. A. Taflove, Computational Electrodynamics: The FDTD Method (Artech House, 1995). R.M. Joseph, S.G. Hagness and A. Taflove, Optics Lett. 18 (1991) 1412. P.M. Goorjian, A. Taflove, R.M. Joseph and S. Hagness, IEEE J. Quantum Electron. 28 (1992) 2416. A. Taflove, Wave Motion (1988) p. 547. R.M. Joseph and P.M. Goorjian. Optics Lett. 48 (1993) 491. S. Radic and N. George, Optics Lett. 19 (1994) 1064. J.G. Blaschak, J. Comput. Phys. 77 (1988) 109. J.P. Berenger, J. Comput. Phys. 114 (1994) 185. B. Engquist and A. Majda, Math. Comput. 31 (1977) 629. N. George, Optics Comm. 133 (1997) 22.