EM Pulse Transit across a Uniform Dielectric Slab J. A. Grzesik NORTHROP GRUMMAN SPACE TECHNOLOGY One Space Park Building R11, Mail Stop 2856 Redondo Beach, CA 90278 U. S. A. telephone: (310) 812-1073 email:
[email protected]
Abstract— A fully time dependent electromagnetic field solution describing the transit of a flat pulse of width c T across a uniform dielectric sandwich is obtained herein by use of a Green’s function technique in the Laplace transform domain. Once the ensuing integral equation has been solved self-consistently within the dielectric sheet, the remainder of field evolution on its exterior follows constructively without any need to cope with boundary conditions or any sort of temporal sequencing on internal pulse bounces. Transform inversion into the temporal plane, relying on a simple development into geometric series at the appropriate steps of analysis, reveals transmitted/reflected pulse trains composed of whole retinues of secondary echo pulses suitably arranged in time and diminished by ascending powers of in/out reflection coefficients. A transcription into computer code verifies that electromagnetic energy content in the transmitted/reflected pulse trains properly sums into that of the impinging pulse, and yields a graphical depiction of the individual, fore/aft pulse trains. Additional work, elsewhere reported, goes on finally to generalize the uniform pulse solution so as to embrace a slab excitation that is profiled [read “carrier modulated by an amplitude/phase signal”] in response to a magnetic amplitude h( τ ) extended across a temporal support of width τ = T past the point of first contact. Index Terms—Electromagnetic transient propagation, electromagnetic transient scattering
I. I NTRODUCTION This presentation undertakes to describe a fully dynamic electromagnetic scenario wherein a flat pulse of width c T impinges upon a uniform, nondispersive and nondissipative dielectric slab of thickness a. Prior to impact, which is taken as the origin of time t , and which occurs along the slab normal, the dielectric material is electromagnetically quiescent, with both electric E and magnetic H fields strictly null. The general outlines of field evolution immediately leap to mind on the basis of a primitive physical expectation that the pulse, following its penetration into the slab, is condemned to endure an endless cascade of back and forth bounces, each diminished by the square of the in/out reflection coefficient. Accompanying this progressively attenuated internal rattling is a reflected/transmitted field leakage into the exterior, one wherein a dominant, reflected/transmitted pulse is trailed by an endless, albeit dwindling retinue. It is just this picture which we seek to erect, and we propose to do so from the vantage point of field self-consistency which
is offered by transcription of Maxwell’s equations into an integral, Green’s function form. And, as the problem is time dependent, we do so only after having invoked a Laplace transformation en masse. It will then quickly emerge that, once a proper analytic foundation has been put into place, the remainder of field solution follows quite smoothly, without any overt need to cope with interface boundary conditions, or even to track the temporal sequence of internal bounces. All such perceived complications are automatically resolved by an apparatus of immense calculational power, one which, following upon transform inversion, leads inexorably to the physical picture just now outlined. The problem thus introduced is of course highly idealized. Such simplification notwithstanding, it would seem to have some relevance to signal detection by instruments of high temporal resolution. But, regardless of its practical motivation, it provides a happy marriage indeed of physical intuition and the workings of analysis in a setting which, while not completely trivial, is nevertheless simple enough to assure total success. II. A G REEN ’ S F UNCTION F IELD A PPARATUS T RANSFORM D OMAIN
IN THE
We take the common direction of slab normal and pulse propagation as the z axis, with the origin so situated that the slab extends from z = 0 to z = a. Accordingly, at the moment of first impact, the pulse extends from z = −c T to z = 0. Within the pulse the magnetic field is taken to have a uniform strength of 1 amp`ere per meter, and to be directed along the y axis. Thus, if we further denote by U + the Heaviside unit step, positive together with its argument, then we have: H i (z) = ˆ ey U+ (z + c T ) U+ (− z ) ,
(1)
with subscript i signifying an initial value. The electric companion E i (z) of (1) has amplitude 1 Z o = (µ o / o ) 2 = 376.73 Ω , and is directed along ˆ ex , with a sense σ = ± 1 responding to a forward/retrograde pulse propagation. Evidently, in the event that σ = − 1 , pulse and slab remain forever electromagnetically disengaged, and it is incumbent upon our succeeding analysis to confirm such field neutrality. We verify these attributes of E i (z) by
requiring that Amp`ere’s law be valid along the forward/aft pulse edge, which amounts here to writing E o for its amplitude and then setting: ∇×{ˆ e y U + ( σ c t − z ) } = o E o ∂ t U + ( σ c t − z ) , (2)
or else: − δ (σ ct − z ) ˆ ez × ˆ ey = σ
r
o E o δ ( σ c t − z ) , (3) µo
whence the stated claim follows as a simple balance among Dirac deltas δ . Altogether then, E i (z) = ˆ e x σ Z o U + ( z + c T ) U +( − z ) .
(4)
The vectorial field nature having thus been fully adjudicated, we henceforth designate the corresponding component values by capital script letters, viz., E ( z , t ) and H ( z , t ) , and, with a view to an imminent development, their Laplace transforms through the simple shorthand of having reference to time t as argument supplanted in favor of transform variable s . And, in general, whenever the enveloping context so permits, we will suppress argument reference in toto so as to streamline the writing. The ability of the dielectric wafer to influence field evolution resides ultimately in the polarization currents induced throughout it, and hence in the amount of radiation which is to be ascribed to them. It is just this physical consideration which lies at the root of the integral equation about to emerge. A gauge of this propensity to radiate is found in the difference, ˜ − o , between the wafer permittivity ˜ , constant and both nondispersive and nodissipative by assumption, and its reference, exterior value o = 8.854 × 10 − 12 farad per meter. The profile of dielectric permittivity ( z ) can thus be taken to read: ( z ) = o + ( ˜ − o ) U + ( z ) U + ( a − z ) .
(5)
Under Laplace transformation the Faraday and Amp`ere equations are brought into the respective forms: ∂z E = µo Hi − µo s H
(6)
∂z H = o Ei − s E ,
(7)
and: whereupon, following elimination of H and a reference to (5) we encounter the wave equation: 2 ∂z − µo o s2 E = µo ∂z Hi − µo o s Ei + (8) µ o s 2 ( ˜ − o ) U + ( z ) U + ( a − z ) E , having the Green’s function: c − s |z −z 0 | e c 2s and hence an integral transcription as: Z ∞ G(z, z0 ) = −
µo o s
0
0
G (z , z ) Ei (z ) d z + −∞
µ o s 2 ( ˜ − o )
Z
a
G(z , z 0 ) E (z 0 , s) dz0 . 0
in consequence of which: Z ∞
(11)
G(z , z 0 ) ∂z 0 Hi (z 0 ) d z 0 =
µo
−∞
−
c µo 2s
s
s
e− c |z+cT | − e− c |z|
whereas (4) similarly leads to: Z ∞
,
(12)
G( z , z 0 ) Ei ( z 0 ) dz 0 =
−µo o s
−∞
σ µo 2
Z
0
s
e− c |z−z
0
|
dz 0 .
(13)
−cT
For the computation of the axial Poynting flux S ( z , t ) = E ( z , t ) H ( z , t ) we shall further require knowledge regarding H ( z , s ) . This latter follows now from (7) as a simple quadrature: Z z 0 0 0 0 H(z , s) =
(z ) Ei (z ) − s E ( z , s) dz
(14)
∓∞
wherein we can, and shall, use either form at will so as to minimize calculational effort in each local circumstance. III. S ELF -C ONSISTENT F IELD S OLUTION I NTERIOR TO THE S CATTERING WAFER We introduce next the refractive index n > 1 by setting = o n 2 . A restriction of (8) to 0 < z < a allows us ˜ then to write: s nz
s nz
+ A− e− c
E ( z , s) = A+ e c
,
(15)
with amplitudes A ± still to be determined. Such determination is made possible by the circumstance that the third quadrature on the right in (10), when confronted by (15), has the remarkable effect of inducing a strict replication (leading to the so-called Ewald-Oseen extinction), accompanied by free-space, exp ( ± sc z ) modes with well defined coefficients linear in the desired amplitudes A ± . And so, as the first and second quadratures in (10) engender nothing other than such free modes, a balance among their net coefficients suffices to fix A ± with full precision. One demonstrates the requisite replication by noting that, apart from various factors, the third quadrature on the right in (10) Z insists that we now consider: s |z−z0 | c
e−
0
−∞ ∞
0
∂z Hi ( z ) = δ ( z + c T ) − δ (z ) ,
a
G( z , z 0 ) ∂z 0 Hi (z 0 ) d z 0 −
E (z , s) = µo
Z
(9)
It is in the nature of this latter equation that it yields all at once a self-consistent field solution within the scattering wafer and an ex post facto field construction throughout its exterior largely as a quadrature afterthought. Of course, in (10) the integrals over initial fields H i and E i , although formally extended full range, allude in fact only to the { − c T , 0 } support of the impinging pulse at the moment of its wafer impact. Indeed, from (1) it follows that:
=
(10)
=
e
− s z c
Z
e±
s nz0 c
z
e
s (1±n)z0 c
0
dz + e
0
c e− s(1 ∓ n) c − e− s(1 ± n)
−
dz0
s (1∓n)a c
s z c
+
e
s z c
s z c
Z
a
e−
s (1∓n)z0 c
dz0
z
2c e± s( 1 − n2 )
(16) s nz c
.
And so, once the deferred factors from (9) and (10) have been duly acknowledged, the last entry bespeaks indeed a full cancellation of the field, as rendered here by (15), from the left-hand side of (10). There remain thus as a diminished legacy of (16) only its free-mode contributions. Pieces (12) and (13) produce naught but free-mode contributions, Z which for z > 0 adopt a common form. Thus: ∞
0
0
G (z , z ) ∂z0 Hi (z ) dz
µo
0
−∞
cµo = 2s
and: −µo o s
Z
1 − e
− sT
∞
e
− s z c
(17)
G(z , z0 ) Ei (z 0 ) dz0
Z
− s (1−n)a c
+ A− ( 1 − n) e
s (1+n)a
∆ = (1 − n )2 e− c
s (1+n)a − c
(19)
= 0 (20)
s
− ( 1 + n ) 2 e − c ( 1 − n ) a (21)
1 − e
s∆
(1 ∓ n) e
, (22)
c
and, in a formal sense at least, our solution is complete. It remains now only to adapt master connection (10) to the various geometric regimes at hand. For this purpose we divide the propagation direction into four regions: Region I: z < − c T ; Region II: − c T < z < 0 ; Region III: 0 < z < a ; and lastly Region IV: a < z . Evidently it is Regions I and IV which are the most pertinent from an observational point of view, and so it is them which we examine first, in reverse order, with Regions II and III considered elsewhere for the sake of completeness. IV. T HE T RANSMITTED P ULSE : z > a For z > a the required analog of (16) reads: Z a s 0 e± c nz
0
= e =
Z
a
e
dz
0
(23)
0
c s(1 ± n)
e
s (1±n)a c
− 1
e−
s z c
.
− 1
s (1−n)a c
− 1
s (1−n)a c
1 − e
(24)
i
e−
s z c
s z c
e−
−sT
s e− c z
1 − e
(1 + n)2
(25) ,
×
s e− c [z −{1−n}a]
s
1 −
1−n 2 1+n
e−2
,
+
c
2 ( 1 + σ ) cµo n s∆
which can at length be even more explicitly rendered as: 2( 1 + σ ) cµo n −sT s na c
.
(26)
We should then hasten to remark that (14), when invoked with its lower sign choice, readily confirms the expectation that: H ( z , s ) = Z o− 1 E ( z , s ) .
(27)
At this point it is well to pause and examine the physical meaning of the various algebraic ingredients participating in (26). Indeed, the obvious invitation to evolve its denominator into a geometric series blends neatly with a realization that β = { ( 1 − n ) / ( 1 + n ) } 2 is nothing other than the square of the in/out reflection coefficient, conveying precisely that amount of field attenuation which attends one back and forth bounce, whereas, within the accompanying exponent, τ o = 2 n a / c is the bounce duration. In similar fashion, in the exponent of the numerator, [ z − { 1 − n } a ] / c conveys the cumulative retardation which the forward pulse edge must hurdle in order to reach point z . And lastly, as ( 1 + σ ) , when not identically zero, is just 2, we recognize the overall factor 4 n / ( 1 + n ) 2 as being the product of {out/in} by {in/out} transmission coefficients appropriate to a default, straight through wafer crossing, without any intervening bounce. Writing thus τ for a generic temporal variable and subtracting from it the aforementioned retardation so as to get ζ = τ − [ z − { 1 − n } a ] / c , we can render field E ( z , t ) as a difference between fore and aft pulse front contributions: E (z , t) = P (z , t) − P (z , t − T ) ,
(28)
wherein P ( z , τ ) (short for “penetrating”) is assembled via transform inversion along a Bromwich contour as: ∞ 2 (1 + σ ) cµo n X P (z , τ ) =
dz0
s (1±n)z0 c
e
s c (1+n)a
A+ (1 − n) e
2
E (z , s) =
for the determinant of system (19) and (20), we find the amplitudes themselves in the form: (1 + σ ) cµo −sT − s (1±n)a
− s z c
E (z , s) =
= −
1 − e
s
e
or else, with a view to (19) and (22): s (1+n)a 1
(18)
fixing wafer amplitudes A ± . Factor ( 1 + σ ) on the righthand side in (19), in particular, makes good at long last on our earlier promise that σ = − 1 (i.e., retrograde pulse propagation) leads to a total pulse/wafer disengagement, circumstance which we henceforth avoid by having in mind only the alternative option σ = 1 , even though, for the sake of generality, we do retain σ = ± 1 as an overt parameter. Writing next:
s 0 e− c |z −z |
1 A+ (1 − n) 2 1 + A− (1 + n) 2 +
0
With (15) through (18) brought beneath the purview of (10) and a balance individually enforced upon the fore/aft free modes, one easily arrives at the following 2-by-2 linear system: ( 1 + σ ) cµo −sT A+ (1 − n) + A− ( 1 + n ) =
1 − e
2s
A− (1 + n) e
s 0 σ µo − s z e c e c z dz0 2 −cT σ cµo s = 1 − e−sT e− c z . 2s
A± = ±
E (z, s) =
−∞
=
A+ (1 + n ) e
Since both (17) and (18) remain intact, we find on the strength of (10) that: h (1 + σ ) cµo −sT
(1 + n)2
β l U+ (ζ − τo l) .
(29)
l = 0
This series, albeit straggling formally to infinity, clearly breaks off at index l > = b ζ / τ o c , and can thus be displayed in analytic form as: ( 1 + σ ) cµo 1+ l P (z, τ ) =
2
1 − β
>
U+ (ζ ) .
(30)
We elsewhere repeat this calculation via a residue sum and its ensuing Fourier series. While somewhat less efficient, this alternative method is not without intrinsic interest, and it does provide us with a check on the present work. Moreover, a companion development reveals that work as being little more than a convenient stepping stone to pulse profiles textured with a magnetic amplitude h( τ ) maintained throughout a temporal interval τ = T past the point of impact. Suitable generalizations are provided then of the analytic triplets (28)-(30) and (33)-(35) respectively applicable to the transmitted/reflected pulse trains. Transcription as code for the work already in hand additionally yields Figures 1 and 2 as illustrations of pulse transmission, and Figures 3 and 4 for its reflected counterpart. In these figures, a 1 ns pulse is allowed to traverse a 1 cm thick wafer of distilled water, n ≈ 9 . Bounce duration τ o = 2 n a / c ≈ 0.6 ns thus comports well with the evidence of Figures 2 and 4. Figure 1. Electromagnetic Pulse Transmission across a Nondispersive, Nondissipative Dielectric Sheet Dielectric Constant = 81.1 (Distilled Water), Wafer Thickness = 1 cm, Pulse Duration = 1 ns
E (z , t) = R(z , t) − R(z , t − T ) , (1 + σ ) cµo R(z , τ ) = 2
"
4n U+ (ξ ) − (1 + n)2
∞
X
β
m
1 − n 1 + n
(33)
×
U+ (ξ − τo {m + 1})
m=0
#
.
(34)
As the series again terminates, this time at index m > = max { b ξ / τ o c − 1 , 0 } , we encounter:
Front Edge Delay = 50.3 ns 0
R(z, τ ) =
-20 Normalized Pulse Power [dB]
with a negative sign befitting retrograde power flow. All in all (31) prods our physical intuition once more by decomposing the reflected pulse into a primary reflection at the z = 0 interface augmented by an endless train of internal bounces. All such internal bounces are of course prefaced with the {out/in} by {in/out} transmission coefficient product, exactly as was found in connection with the transmitted pulse, and they exhibit also an overall minus sign which alone differentiates between the {out/out} and {in/in} reflection coefficients. We may next emulate the time domain recovery (28) through (30) by setting ξ = τ + z / c as our new retarded variable and writing ( R being short for “reflected”):
-40 -60
(1 + σ )c µo 2
U+ (ξ ) −
1 − β
1 − n 1 + n
1+ m>
as the closed form analogue to (30).
-80
×
U+ (ξ − τo )
(35)
Figure 3. Electromagnetic Pulse Reflection from a Nondispersive, Nondissipative Dielectric Sheet -100
Dielectric Constant = 81.1 (Distilled Water), Wafer Thickness = 1 cm, Pulse Duration = 1 ns Front Edge Delay = 51.7 ns
-120 0 900
1000
1100 1200 1300 Pulse Location [cm]
1400
1500
1600
-20
Figure 2. Electromagnetic Pulse Transmission across a Nondispersive, Nondissipative Dielectric Sheet Dielectric Constant = 81.1 (Distilled Water), Wafer Thickness = 1 cm, Pulse Duration = 1 ns Front Edge Location = 1500 cm 0
Normalized Pulse Power [dB]
-20
Normalized Pulse Power [dB]
-140 800
-40 -60 -80 -100 -120
-40 -140 -1600
-60
-1500
-1400
-1300
-1200
-1100
-1000
-900
-800
Pulse Location [cm]
-80 Figure 4. Electromagnetic Pulse Reflection from a Nondispersive, Nondissipative Dielectric Sheet Dielectric Constant = 81.1 (Distilled Water), Wafer Thickness = 1 cm, Pulse Duration = 1 ns
-100
Front Edge Location = -1550 cm -120
0
-140 50
55
60
65
70
-20
75
Pulse Observation Time [ns]
V. T HE R EFLECTED P ULSE : z < − c T Adaptation of (9) and (10) to the echo domain with z < − c T follows similar lines and yields E (z, s) =
( 1 + σ )c µo 2s
1 −
4n (1 + n)2
1 − n 1 + n
1 − e−sT
e−τo s 1 − β e− τo s
e
s z c
×
(31)
as the analogue of (26). A further appeal to (14), but now with its upper sign option exercised, produces at once: H ( z , s ) = − Z o− 1 E ( z , s )
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(32)
Normalized Pulse Power [dB]
45
-40 -60 -80 -100 -120 -140 50
55
60
65
70
75
Pulse Observation Time [ns]
VI. R EFERENCES 1. J. A. Grzesik and S.-C. Lee, “The dielectric half space as a test bed for transform methods,” Radio Science, Volume 30, Number 4, Pages 853-862, July-August 1995.
