IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. 2, MAY 1989 197

Correspondence

Comments on “Solutions of Maxwell’s Equations for General Nonperiodic Waves in Lossy Media’’

TERENCE W. BARRETT

The intention of El-Shandwily’s paper [ 11 is to demonstrate that the linear Maxwell’s equations can be applied to the case of a pulse or step-function field change. If this can be demonstrated, then, it is claimed, the Harmuth Ansatz [2]-[4] for solving Maxwell’s equations to obtain the magnetic field for impulse excitation is unnecessary. In this letter, I show that El-Shandwily actually implements the Harmuth Ansatz and this explains the concurrence of Harmuth’s and El- Shandwily’s predictions.

In El-Shandwily’s paper, the solutions of the magnetic field strength H(.$‘, 0) [ l , eqs. (18) and (20)] are derived from the integrals [ l , eqs. (15) and (16)]

a f / a . $ ‘ [ 2 ~ ( e - .$‘)/a] de = - { df ( x ) / d x } dx L O

= - f ~ 4 0 - ~1 + f m (2)

where x = 26(0 - .$‘)/a, p is permeability, e is permittivity, a is conductivity, 0 = [ a / 2 ~ ] t , .$‘ = [ (a /2 ) (@e) ] z , and E(.$‘, 8) is the electric field strength due to an exciting function f ( 2 ~ ( 0 - .$‘)/a).

Firstly, we note that the derivative in (2) is with respect to a space variable .$‘, while the integral is over the time variable 8. Therefore, the first assumption (covert or overt) or boundary condition of El- Shandwily is

[U/2€] t = [ (U/2 ) ( ‘&/€ ) (3)

or

t = J;;z. (4)

One of the implications of this is that

Then there is the second assumption

f ( 2 4 e - . $ ‘ ) / U ) = 6(24e - .$‘)/a) (6)

which, as has been pointed out in letters by Harmuth [5] and Hussain [6] , amounts to equating a distribution rule with a function.

What do these two assumptions mean physically? The first

Manuscript received November 12, 1988. The author is with Boeing Aerospace, Arlington, VA 22209. Tel. (703)

IEEE Log Number 8926408. 558-3288.

assumption centers the discussion at t = 0 and z = 0. Also, as stated by the author, f(0) = 0. This assumption is, of -ourse, valid for the U(1) symmetry, linear, or Abelian Maxwell’s equations. But if this were all that existed in the paper, we need proceed no further, because there would be no signal to propagate. However, the second assumption permits the discussion to continue but not, as El- Shandwily claims, in the linear theory, but in the SU(2) symmetry, nonlinear, or non-Abelian Maxwell theory. Why this is so is as follows.

Let us consider the second (mathematically incorrect) assumption in depth. A distribution rule that acts on a function (of, for example, n degrees of freedom in transformation rule symmetry) has more degrees of freedom than that function (at least n + 1). However, to equate a distribution rule with a function means, physically, that the distribution rule and function have the same number of degrees of freedom in transformation rule symmetries. In the case of e.m. fields, this is equivalent to requiring the function to be of SU(2) symmetry, matching the symmetry of the distribution rule.

Is, then, the (6) equivalence a plausible equivalence? No, not in the case of an equivalence, but if we treat the assumption as if attempting to state that the energy function is unaffected (or almost so) by application of the distribution rule, then the answer is yes. Classical aperture theory is the methodology of determining e.m. fields given the appropriate aperture field distrubtion, and linear transduction is what occurs when a step function input is transduced with no (or little) dispersion and nonlinearity by a lens, or when an impulse is transduced by an (approximately) frequency-independent antenna. Another example is of an impulse exiting without dispersion and nonlinearity into our U(1) symmetry world from a cavity of SU(2) symmetry. Such examples indicate that there exist situations where rules for the distribution of energy do not affect, change, or condition precisely matched energy functions. However, to ask for equiva- lence amounts to equating operator and operand and to ignore the required matching of the energy function, i.e., the special nature of those situations.

Can one make this second assumption formally? Yes, formally one may use the double-Wigner distribution function (or Wigner-Ville distribution [7]-[ 111

. .

fO(t0, 00)=(1 /2~) J K( to , mor ti, wi)f , ( t i , wi) dti dui (7)

where the subscripts i and o signify input and output, K is the time- frequency domain response of the system, or a ray spread function, and f( t , w ) is defined with respect to input-output relations as

f ( t , w)= 1 cp(t+(l /2) t ’ )cp*(t-(1/2) t ’ ) exp [ - i w t ’ ] d t ’ (8)

where (p is a signal. Then El-Shandwily’s second assumption amounts to

(9)

which is the dual of a single ray or sinusoid, i.e., an impulse,

0018-9375/89/0500-0197$01.~ 0 1989 IEEE

1 9 8 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 3 1, NO. 2, MAY 1989

E(1-IN)

E(2-OUT)

(instantaneous bandwidth) launch over a wide-band approaching zero 4 7-L dispersion in nonlinearity.

E(2-IN) In a practical sense, the Booker antenna relation [15]

where Zd is the impedance of a dipole antenna and Zs is the impedance of its complementary slot antenna, addresses the possibil- ity of transduction without dispersion or nonlinearity (wide-band, parallel-frequency launch). That is the origin of the interest in frequency-independent antennas by impulse radar researchers [ 161- [19]. For example, after photoactivated switches have created the

E(2-OUT1

n E(1-IN1

(b) I E(2-1N)

E(l-IN1, E(1-OUT)

conditions for an e.m. impulse to be transmitted, that impulse must be launched (transmitted) by the antenna system without distortion or nonlinearity to exploit the possibility of an ultrawide-band signal. However, Booker’s relation is only exact for idealized slot and dipoles of zero thickness that symmetrically radiate equally well in both sides of an aperture [20]. There is no finite-sized antenna that is really frequency-independent over the frequency range 0 to infinity, but the possibility is always there that by increasing the size one may approximate more exactly the ideal. Similarly, El-Shandwily ’s hidden requirement for linear and dispersionless transduction of an impulse (his second assumption) is a worthy requirement, but nevertheless can only be approximated. It requires, as noted above,

-b

(C) Fig. 1 . SU(2) field conditioning interferometers: (a) Fabv-Perot; (b)

Mach-Zehnder; (c) Stokes.

parallel frequency (instantaneous bandwidth) launch, which is a

rules. transduction described by SU(2) symmetry energy conditioning

entering the system at time to with all frequencies present. It must be emphasized that we are dealing with an impulse, which is the dual of a sinusoid, in this assumption. Let us consider this in the theory of diffraction. A lens that can transduce all frequencies with no dispersion and nonlinearity is a lens with no focal point, i.e., no lens at all. Theoretically, however, dual forms can be treated in SU(2) symmetry, a physical representation of such transformation rules of symmetry being an impulse source at the focal point of an infinitely large parabolic reflector. The problem is in dealing with dual forms, one parameter of which involves an infinite measure.

Cavity waveguide interferometers restore symmetry (increase degrees of freedom in the transformation rules) and examples are shown in Fig. 1 . The Mach-Zehnder and the Fabry-Perot are SU(2) conditioning interferometers [ 121. The third shown is a polarization modulator [13] that also restores SU(2) symmetry, and that we called the Stokes’ interferometer [13], [14]. The SU(2) group characterizes passive lossless devices with two inputs. If such devices are excited by a matched impulse and that impulse exists without dispersion or amplitude modulation (nonlinearity) (which can be achieved only approximately), then one might claim that application of the distribution rule (of the cavity) to the e.m. function (the input) resulted in a transduction that did not perturb that matched impulse, i.e., the pulse was transduced linearly. Cavities with higher-order symmetries are, of course, possible, and matched e.m. functions will be transduced by them without dispersion or nonlinearity. We do not say, however, that the distribution rule or energy conditioning is the same as the field it acts on. We say, rather, that a matched function acquired neither dispersion nor nonlinearity (a.m. modulation) when acted upon by the distribution rule, i.e., when conditioned by the cavity.

Another well-known endeavor to achieve dispersionless and linear transduction through a conditioning medium is the search for the frequency-independent antenna. Such an antenna is required for dispersionless and linear launch of impulses preserving bandwidth. This requirement is not to be confused with medium band antennas for FM signals with the frequencies launched serially low-to-high, or high-to-low. It is, rather, the requirement for parallel frequency

As an aside, it may also be pointed out that 1) a distribution rule- energy function match (not equality) providing linear and dispersion- less transduction, and approached by parallel frequency transduction, is not the same as 2) matching the distribution rule (this time not a Dirac delta function rule, but another rule) to an imperfect transducer (i.e., with nonlinearity and dispersion, e.g., a cylindrical waveguide, distributed apertures, a parabolic reflector). The latter occurs in the launch of Brittingham waves (21) and wave varieties thereof [22]- [MI. The transducing requirements for these latter waves 2) are different from the former waves 1) under discussion here.

In conclusion, El-Shandwily unintentionally makes two covert assumptions that amount to invoking the same physical conditions required by the Harmuth Ansatz, namely, the assumption of SU(2) symmetry conditioning of the e.m. field. This was pointed out in a previous letter [45] and papers [13], [14], [46]. Simply stated: The linear transduction of dual forms involving infinite measures can only be performed by transducers of SU(2) symmetry transformation rules. This prevents Abelian Maxwell theory from describing the complete process. The adoption of the two assumptions by El- Shandwily is tantamount to the adoption of SU(2) symmetry rules, i.e., non-Abelian Maxwell theory.

It is entirely to be expected, therefore, that El-Shandwily’s predictions [ l , Fig. I] agree with Harmuth’s. However, the analysis presented here indicates that El-Shandwily ’s conclusions that: 1) the linear Abelian Maxwell theory is appropriate for the treatment of impulse responses; and 2) Harmuth’s use of non-Abelian Maxwell theory (under the appropriate circumstances) is unnecessary, are incorrect.

This is thus a case of “plus p change, plus la mEme chose.” The Harmuth Ansatz is unintentionally implemented by El-Shandwily and is necessary for the solution of Maxwell’s equations in the situation of impulse signals propagating through lossy (nonlinear) media.

REFERENCES [l] M. E., El-Shandwily, “Solutions of Maxwell’s equations for general

nonperiodic waves in lossy media” IEEE Trans. Electromagn. Cornpat., vol. 30, no. 4, pp. 577-582, Nov. 1988. H. F. H m u t h , “Correction of Maxwell equations for signals I,” [2]

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 31, NO. 2, MAY 1989

I

199

ZEEE Trans. Electromagn. Compat., vol. EMC-28, no. 4, pp. 250- 258, Nov. 1986. -, “Correction of Maxwell equations for signals II,” ZEEE Trans. Electromagn. Compat., vol. EMC-28, no. 4, pp. 267-272, Nov. 1986. -, ‘‘Propagation velocity of electromagnetic signals,” ZEEE Trans. Electromagn. Compat., vol. EMC-28, no. 4, pp. 267-272, Nov. 1986. -, “Comments on a short paper by El-Shandwily on transient solutions of Maxwell’s equations,” ZEEE Trans. Electromagn. Compat., vol. 31, no. 2, pp. 199-200, May 1989. M. G. M. Hussain, “Comments on ‘Solutions of Maxwell’s equations for general nonperiodic waves in lossy media,” IEEE Trans. Electromagn. Compat., vol. 31, no. 2, pp. 202-204, May 1989. J. Ville, “Theorie et application de la notion de signal analytique,” Cables et Transmissions, vol. 2 , pp. 61-74, 1948. B. Bouachache, “Sur une condition necessaire et suffisante de positivitk de la representation conjointe en temps et frequence des signaux d’energie finie,” Compte Rendus Acad. Sciences, vol. 288, siere A, pp. 307-309, Jan. 1979. -, “Representation temps-frequence,” Thes. Doct. Ing., Inst. Nat. Polytechnique, Universitk de Grenoble, May 1982. B. Boashash, B. Black, and H. J. Whitehouse, “An efficient implementation for real time application of the Wigner-Vale distribu- tion,” in Real Time Signal Processing, SPIE Optical Information Processing Conf. (San Diego), pp. 262-268, Aug. 1986. B. Boashash, “Non-stationary signals and the Wigner-Ville distribu- tions: a review,” in Proc. ZSSPA87, Signal Processing: Theories, Implementations and Applications (Brisbane, Australia), vol. 1,

B. Yurke, S. L. McCall, and J. R. Klauder “SU(2) and SU(1, 1) interferometers,” Phys. Rev., vol. A33, no. 6, pp. 40434054, June 1986. T. W. Barrelt, “Maxwell’s theory extended,” unpublished manuscript, 1988. -, “Energy transfer dynamics: Signal SU(2) symmetry condition- ing for radiation, propagation, discrimination and ranging,” A. J. Schafer Assoc., Inc., Rep., June 1987. H. G. Booker, “Lot aerials and their relation to complementary wire aerials (Babinet’s Principle),” J. Inst. Elect. Engrs., part 11 IA, vol. 93, no. 4, pp. 620-6, 1946. R. H. DuHamel and G. G. Chadwick, “Frequency-independent antennas,” in R. C. Johnson and H. Jasik, (eds), Antenna Engineering Handbook. V. H. Rumsey, Frequency Independent Antennas. New York: Academic, 1966. H. F. Harmuth, Nonsinusoidal Waves for Radar and Radio Communication. New York: Academic, 1981. -, Antennas and Waveguides for Nonsinuoidal Waves. New York: Academic, 1984. D. R. Rhodes, Synthesis of Planar Antenna Sources. Oxford: Claredon, 1974. J. N. Brittingham, “Focus modes in homogeneous Maxwell’s equa- tions: Transverse electric mode,” J. Appl. Phys., vol. 54, no. 3, pp.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett., vol. 7, pp. 684-685, 1971. P. A. slanger, “Packetlike solutions of the homogeneous wave equation,” J. Opt. Soc. Amer., A, vol. 1, no. 7, pp. 723-724, July 1984.

Aug. 24-28, 1987.

New York: McGraw-Hill, 1961, pp. 14-1-1443.

1179-1189, Mar. 1983.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling wave representation of exact solutions of the scalar wave equations,” J. Math. Phys., in press, 1988. A. M. Shaarawi, I. M. Besieris, and R. W. Ziolkowski, “Localized energy pulse trains launched from an open, semi-infinite circular waveguide,” J. Appl. Phys., in press, 1988. J. Durnin, “Exact solution for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Amer., A, vol. 4, no. 4, pp. 651-654, Apr. 1987. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett., vol. 58, no. 15, pp. 1499-1501, Apr. 1987. T. T. Wu, “Electromagnetic missiles,” J. Appl. Phys., vol. 57, no. 7, pp. 2370-2373, Apr. 1985. T. T. Wu, R. W. P. King, and H-M. Shen, “Spherical lens as a launcher of electromagnetic missiles,” J . Appl. Phys., vol. 62, no. 10, pp. 4036-4040, Nov. 1987. T. T. Wu and H-M. Shen, “Radiation of an electromagnetic pulse from the open end of a circular waveguide,” in Microwave and Particle Beam Sources and Propagation, Proc. SPZE (Los Angeles, CA),

H-M. Shen, “Experimental study of electromagnetic missiles,” in Microwave and Particle Beam Sources and Propagation, Proc. SPZE (Los Angeles, CA), vol. 873, pp. 338-346, Jan. 1988. J. M. Myers, “Pulsed radiation from a parabolic reflector,” in Microwave and Particle Beam Sources and Propagation, Proc. SPZE (Los Angeles, CA), vol. 873, pp. 347-356, Jan. 1988. S . Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Sci. Amer., vol. 67, no. 5, pp. 699-700, May 1977. L. B. Felsen, “Real spectra, complex spectra, compact spectra,” J. Opt. Soc. Amer., A, vol. 3, no. 4, pp. 486-496, Apr. 1986. E. Hyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,’’ J. Opt. Soc. Amer., A, vol. 4, no. 11, pp. 2081- 2091, Nov. 1987. E. Heyman and B. Z. Steinberg, “Spectral analysis of complex-source pulsed beams,’’ J. Opt. Soc. Amer., vol. 4, no. 3, pp. 473-480, Mar. 1987. E. Heyman and L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,” ZEEE Trans. Antennas Propagat., vol. AP-34, no. 8, pp. 1062-1065, Aug. 1986. L. B. Felsen and E. Hyman, “Discretized beam methods for focused radiation from distributed apertures,” in Microwave and Particle Beam Sources and Propagation, Proc. SPZE (Los Angeles, CA), vol. 873, pp. 320-328, Jan. 1988. T. W. Barrett, “Comments on the Harmuth Ansatz: Use of a magnetic current density in the calculaton of the propagation velocity of signals by amended Maxwell theory,” ZEEE Trans. Electromagn. Compat., vol. 30, no. 3, pp. 419-420, Aug. 1988. -, “On the distinction between fields and their metric: the fundamental difference between specifications concerning medium- independent fields and constitutive specifications concerning relations to the medium in which they exist,” Annales de la Foundation Louis de Broglie, in press, 1988.

vol. 873, pp. 329-337, Jan. 1988.

Comments on a Short Paper by El-Shandwily on .. . ..

P. Hillion, “Spinor focus wave modes,” J. Math. Phys., vol. 28, no. Transient Solutions of Maxwell’s Equations 8, pp. 1743-1748, Aug. 1987. _ _ -, “More on focuswave modes in Maxwell’s equations,” J. Appl. Phys., vol. 60, no. 8, pp. 2981-2982, Oct. 1986. HENNMG F. HARMUTH, MEMBER, IEEE

A. Sezginer, “A general formulation Of focus wave modes,’’ J. Appl. Phys., vol. 57, no. 3, pp. 678-683, Feb. 1985. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys., vol. 26, no. 4, pp. 861-863, Apr. 1985. century. -, “New electromagnetic directed energy pulses” in Microwave and Particle Beam Sources and Propagation, Proc. SPZE, (Los Angeles, CA), vol. 873, pp. 312-319, Jan. 1988. -, “Localized transmission of electromagnetic energy,” Phys. Rev., A, in press.

El-Shandwfiy claims that his results are obtained “by straightfor-

in almost half a ward mathematical manipulations” of equations known since 1941

Let us see no One published his

Manuscript received December 2, 1988. The author is with the Deparunent of Electrical Engineering, The Catholic

IEEE Log Number 8926409. University of America, Washington, DC 20064. Tel. (202) 635-5193.

0018-9375/89/0500-0199$01.00 0 1989 IEEE