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NATL INST OF STAND & TECH ST

PUBLICATIONS IIIIIIIIHI lllllllllllllllllliiiiiiiiiii"iTMil" PUBLICATIONS

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* NBS TECHNICAL NOTE 1202

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

.U5753

No. 1202

1985

c. 2
Tm he National Bureau of Standards' was established by an act of Congress on March 3, 1901. The M Bureau's overall goal is to strengthen and advance the nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a

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: Some divisions within the center are located at Boulder, CO 80303. 'Located at Boulder, CO, with some elements at Gaithersburg, MD.
JWITIONAL BUREAU OF STANDARDS

LIBRARY Transient Analysis of Electromagnetic

Reflection From Dispersive Materials ^*J*

A. George Lieberman

Center for Manufacturing Engineering

National Engineering Laboratory

National Bureau of Standards

Gaithersburg, MD 20899

-* *. • %

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldnge, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

Issued March 1985
National Bureau of Standards Technical Note 1202

Natl. Bur. Stand. (U.S.), Tech. Note 1202, 70 pages (March 1985)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1985

For sale by the Superintendent of Documents. U S Government Printing Office, Washington. DC 20402
TRANSIENT ANALYSIS OF ELECTROMAGNETIC

REFLECTION FROM DISPERSIVE MATERIALS

A. George Lieberman

National Bureau of Standards

Washington, DC 20234

ABSTRACT

Theoretical expressions are presented describing the

transient and steady-state temporal evolution of an impulsive,

TE-polarized, plane wave reflected into vacuum from any of a

variety of frequency—dispersive material surfaces. Polar

dielectrics, non-polar dielectrics, metals and plasmas are

treated using, respectively, the Debye, Lorentz, and Drude

material models to investigate the effects of dispersion upon

dimensional measurements performed with optical pulses of

extremely short duration. The more general problem, concerning

the reflection of an optical pulse of any specified waveform, is

resolved by performing a convolution of the incident wave at the

material surface with the previously determined reflection of an

impulsive wave.

Key words: Debye dielectric; dispersion, Drude metal}

electromagnetic fields; Green's function; impulse response;

Lorentz dielectric; material models; optical pulses* plasma;

reflected waves; transient fields; waveform analysis.

iii
CONTENTS

Pagi

1. INTRODUCTION 1

2. DIELECTRIC THEORY 3

2.1 Terminology 3

2.2 The Lorentz Dielectric 7

2.3 The Debye Dielectric 16

3. PLANE WAVE PROPAGATION AND REFLECTION 22

4. INTEGRAL TRANSFORMATIONS AND THE FRESNEL THEORY 30

5. CONSTRUCTION OF THE TRANSIENT PROPAGATOR 37

6. THE TRANSIENT PROPAGATOR FOR TREATING REFLECTIONS

FROM A LORENTZ DIELECTRIC 40

7. THE TRANSIENT PROPAGATOR FOR TREATING REFLECTIONS

FROM A DRUDE METAL AND A PLASMA 50

8. THE TRANSIENT PROPAGATOR FOR TREATING REFLECTIONS

FROM A DEBYE DIELECTRIC 55

9. SUMMARY AND CONCLUSIONS 60

10. ACKNOWLEDGEMENT 63

1 1

.

REFERENCES 64

xv
1. INTRODUCTION

A great deal of effort has been expended in understanding

the harmonic time-dependent solutions of Maxwell's equations.

This has happened for a number of good reasons, but primarily

because Fourier's integral theorem permits a source of fairly

arbitary time dependence to be pictured as a superposition of

harmonic oscillators. If the field intensities remain moderate

the radiation from these oscillators can be treated

independently, and if the spectral width of the original source

is sufficiently limited and removed from the resonances of the

supporting medium then all harmonics will propagate with the same

phase velocity. Under these conditions the medium is said to

behave non-dispersi vely and the time dependence of the radiated

waveform at a distant location will reproduce that of the source.

With the recent advent of laser sources capable of

generating femtosecond pulses C13 (containing only a few cycles

of oscillation), and with the expectation of generating still

shorter pulses in the future, the dispersive properties of a

medium can now hardly be ignored. With this in mind, a study of

the transient and steady-state aspects of electromagnetic pulse

reflection from dispersive materials was undertaken.

A review of the archival literature disclosed a considerable

volume of research treating the Q_rgp.agation of optical pulses in

dispersive media. This research commences with the theoretical

demonstration by Sommerfeld and Brillouin C2D that the velocity

of a signal cannot exceed the velocity of light in a region of

Numbers in brackets indicate literature references listed at

the end of this report.
anomalous dispersion. In contrast, the problem of optical pulse

C^£i^£ti.QQ "from dispersive media has hardly been explored. The

little that has been published deals, on one hand, with

dissipative materials for which the constitutive parameters do

not depend upon frequency C3,43. Dispersion in that unrealistic

case is caused by relative changes between the frequencydependent displacement current and the frequency—independent

conduction current. On the other hand, the reflection of radio

frequency pulses from the ionosphere C5-9D has received much more

attention but its applicability to optics is diminished by the

frequency range and material models which have been studied. The

results reported in this paper include the above as special

cases. The material models devised by Debye C103, Lorentz CUD,

and Drude C123, on the basis of classical electron theory, are

employed here to study optical pulse relections from dielectric,

metal and plasma surfaces.
2. DIELECTRIC THEORY

2. 1 Terminology

The behavior of electromagnetic radiation is governed by the

Maxwell equations, the boundary conditions on the -field vectors,

and the constitutive relations describing the supporting medium.

When, as in the present case, the interatomic material forces are

considered short-ranged or weak the constitutive relations assume

a local spatial character and also become independent of the

boundary conditions. Such a medium may, nevertheless, exhibit a

global temporal character in the sense of retaining a memory of

its earlier history, including initial conditions. This

situation arises whenever the relaxation times characterizing the

medium are comparable to the period of the radiation, a condition

satisfied, for example, in an optical field by the motion of

valence electrons in atoms.

For a linear, isotropic dielectric with memory the present

value of the polarization field P(t) is determined by an integral

over all previous values of the impressed electric field E(t):

?ct> =

J

6 X(t-T)E(T)dT. (2-1)

o

n

Both P(t) and E(t) are, of course, to be evaluated at the same

spatial point due to the local character of the medium.

The kernel, £ X(t), of integral equation <2-l) has special

significance in that it describes the response of the dielectric

to an elementary impulsive electric field. Certainly, if the

magnitude of the electric field is given by E(t) = S(t) f where
S (t) is the Dirac delta function, then the polarization -field is

given by P(t) = £ X(t). Function X(t) also has the general

property that it is finite for all values of time, including t=0,

and tends to zero as t^w. The latter simply expresses the fact

that the polarization cannot be appreciably affected by values

of the electric field at sufficiently remote times. In fact,

X(t) differs from zero only for periods of the order of the

relaxation times characterizing the underlying molecular

polarization processes. All materials possess some degree of

memory; the hysteresis exhibited by magnetically polarized iron

is a dramatic example.

Equation (2-1) has the form of a convolution integral in the

time domain. However, under a Fourier transformation, defined by

the pair of integrals:

P(t) = ^ J P(w)e J dw <2-2a)

P(w) = I P(t)e J dt, (2-2b)

— oj

eq. (2-1) can be mapped into an algebraic relation with frequency

w as the independent variable. Namely,

P(w) = e X(w)E(m>, (2-3)

where E(w) and X (w) correspond, respectively, to the Fourier
-* 2 trans-forms of E(t) and X(t). The function X (w) , relating the

dielectric polarization to the electric intensity at each

frequency, is called the dielectric susceptibility while its

dependence on frequency is referred to as dispersion. A material

possessing a strong memory is also highly dispersive.

Conversely, a non-dispersive material (for which X(w)=X is a

constant) possesses no memory at all; that is to say,

X(t)=X * (t)

o

In either representation (time or frequency) the electric

displacement field D is given by

D = € E + P. (2-4)

o

As usual, e is the dielectric constant of vacuum and P describes

o

the effect of the material occupying that physical space. The

advantage of the frequency representation becomes evident upon

substituting eq. (2-3) into eq. (2-4):

D(w) = € Cl+X(w)3E(w) = e(w)E(«). (2-5)

o

The factor € (to) relating D(w) with E(w> is referred to as the

dielectric permittivity of the medium. It is important to again

emphasize that, unless the dielectric susceptibility is non2 Although the same symbol is used to denote the function and its

transform this should not be misconstrued to mean that the^two

functions are ideqtical except for thei r^arguments. Thus P(w) is

not the function P(t) evaluated at t=«; P(t) i.s the Fourier

transform of P(w).
dispersive, the multiplicative structure of constitutive relation

(2—5) is valid only in the -frequency representation.

An expression completely analogous to eq. (2-5) describes

the magnetization state of the given medium. Namely,

B(w) = u (w)H(cj). (2-6)

However, for the vast majority of optical phenomena the

dispersion attributed to magnetization can be ignored because

magnetic resonances occur far below the optical frequency band.

As a result, the constitutive relation in the optical frequency

range joining the magnetic flux density B(o) to the magnetic

field intensity H(w) is usually written

B(w) = u H(m> , (2-7)

o

where u is a constant, often the permeability of vacuum.

The total current flowing through the medium consists of two

components: a conduction current J (w) and a displacement

-+ current jwD(w). The former is described by Ohm's law:

J(w) = ff (u)E(w)

.

(2-8)

The latter, according to eq. (2-5), is also proportional to E(to)

and a convenient practice has been to combine the two components

into a single current by introducing an effective permittivity

function e . _(u). ef f
J(w> +jwD(u) = (<r+jM«)E(u) = jw« <w)E(w). (2-9)

The dispersion of a dielectric medium in the optical frequency

range is therefore fully characterized by its effective

permittivity function. The structure of this function depends

upon the physical model describing the polarization process. The

fundamental processes in dielectric polarization are: (1)

distortion of atomic or molecular charge distributions by an

electric field, and (2) alignment of dipolar molecules along the

local electric field direction. These processes are described,

respectively, by the Lorentz dielectric model and the Debye

dielectric model.

2.2 The Lorentz Dielectric Model

Lorentz pictured dielectric polarization as resulting from

the separation of positive and negative charges in the atomic

structure of a substance by the application of an external

electric field E11JI. For excitations at optical frequencies,

this polarization is ascribed to the dipole moment induced by the

displacement of the valence electrons' center of charge relative

to that of its atomic core. For small displacements the

electrostatic restoring force within an atom is linearly

proportional to the relative displacement, with the result that

the described motion is harmonic. Since the mass of an atomic

core is at least three orders of magnitude larger than that of an

electron, the relative displacement r(t) can be attributed

entirely to the electronic motion which is governed by the

equation:
2-» -» d r dr " -» m hL-= + vm 37- + icr = -eE' <t) . (2-10)

dt 2 dt

The first term represents the inertial reaction of the electron

to its acceleration; the next term describes the viscous damping

of its motion resulting from dissipative interactions (inelastic

collisions) with other particles; the last term on the left deals

with the electrostatic restoring force. Together these three

terms balance the electric force which causes the electron's

d i sp 1 acement

For k=0 this equation describes the motion of a free

electron in an electric field, e.g. a conduction electron in a

lossy metal or plasma. Note that if the electric field is

suddenly removed at time t=0 the equation of motion with k=0

reduces to

m **(£)- o- d_

r

As a consequence, the velocity of the conduction electron is

viscously damped to a standstill with the damping rate determined

by the relaxation time t=1/v of the medium:

dr /dr\ -t/r /r> f _. 5t " Vdt) t=0

e

• <2" 12>

On the other hand, if the applied field is varied

si nusoi dally with frequency w, and expressed as the real part of

8
a complex -field, viz.,

!'<t> = Re g- <.>.*'"*. <2" 13>

then the -forced motion is oscillatory, having an amplitude given

by

-»,.. D

» , % +jwt (2-14) r(t) = Re r(w)e J

Substitution of these expressions into equation of motion (2-10)

now yields:

-> . -<e/m) E' (m) ,_ ,_, r(w) = - -—

,

(2-15)

( jw) J +( jo) J v+w

o

where w =Vic/m is the natural frequency of the oscillator. This

displacement of the electron from its equilibrium position, in

turn, produces an induced dipole having electric moment

-»

,

-»

, (2-16)

p (w) = -er (w)

If N such oscillators Are present in a unit volume of this

dielectric, then the polarization field is given by

-» -» w

2

P(w) = -Ner(w) = e —-—

^ E' (w)

,

(2-17)

( (0 -W ) +j MV

O

r~2

where to =VNe /m« is the electron plasma frequency. The physical

significance of u will become apparent when the dispersion of a

metal or plasma is examined.
More generally, a dielectric could consist o-f several

different kinds of polarizable oscillators each classified by

a particular elastic constant, effective mass, charge, relaxation

time, or density. The total polarization field, in this case,

would be determined as a superposition of the various component

polarizations:

2

w P(w) = e y — &P E' (w) . O L, . Z Z. , . (2-18)

(tO —M )+jWV

n on n

The summation is carried out over all polarization groups; each

denoted by a different value of the subscript n. This expression

can be written also in the form

P(w) = e Y N a (w)E' (w) (2-19)

o n n

n

or, for a single group consisting of N identical atoms

(oscillators) per unit volume, as

P(w) = e Na(to)E'(w), (2-20)

where a(w) is the polar! zabi 1 i ty of a single atom.

It cannot be assumed that the local field E' (w) acting on an

atom is the same as the macroscopic field E(w) applied to the

medium. There is also that electric force exerted by each atom

upon every other atom of the dielectric. In the absence of an

applied electric field these forces combine to create a charge

distribution for which the volume polarization is usually zero.

10
The application of an external field causes each atom to acquire

a new polarization and to exert additional forces on every other

atom. The polarization field within a given volume element now

depends not only upon the direct action of the applied field but

also upon its indirect action through the polarization it creates

in other volume elements. For an isotropic material H. A.

Lorentz C143 and L. Lorenz C153 have shown that the polarization

-+ contributes an amount P/3£ to the electric field, i.e.,

o

E' (to) = E(w) + P(w)/3<s . (2-21)

o

This correction is also valid at a lattice point within a cubic

crystal, but it is not valid for crystals of lower symmetry. For

greater generality relation (2-21) is expressed in the form

E' (w) = E(w) + KP(w)/fi , (2-22)

where the proportionality constant K is to be appropriately

determined for a given medium.

Substituting eq. (2-22) for the local field into eq. (2-19)

and solving for the polarization gives

< X N a (w) n i jtt 1-K I N a (w)

_j o n n P(w) = ~ E(w). (2-23)

n n

n

However, according to eq. (2-3),

11
-+ - P(w) = € X(w)E(w), (2-3)

so that the dielectric susceptibility

I N a (w) n n X(w) = —2—= . (2-24)

1-K L N a (w) n n

n

On comparing eqs. (2-18), (2-19), and (2-20) it appears that

«

2 /N

a (w) = —-—^—- . (2-25)

n . 2 2 . (w —w ) + jw on J n

Within a frequency range determined by the applied -Field,

only a finite number of resonance terms are needed from eq.

(2-24) to describe the polarization state. This being the case,

2

X (w) is a rational function of w allowing eq. (2-24) to be

expressed in the form:

(w -w ) + jUV'

(2-26)

n on n

For example, when a single resonance term suffices, eq. (2—24)

with the aid of eq. (2-25) reduces to

12
2

w X(w) = - 1 ^ (2-27)

(wop—Kw ) —w +jwv

A comparison of this expression with eq. (2-26) shows that

,2 2

P P

W

' 2 = w

2-Kw 2 (2-28)

o op

The e-ffect of the Lorentz-Lorenz correction to the electric field

has been to shift the parameters of the Lorentz dielectric model.

Since the value of K is not always known, the parameters of eq.

(2-26), not those of eq. (2-18), are determined by measurement.

For notational brevity the primes appearing on w' , w' , and v'

will hereafter be suppressed. Thus in place of (2-26) one has

2

X(w) = ) -—Jjp

. (2-29)

no(w —u )+jwv J n

It should be pointed out that when eq. (2-29) is expressed

in terms of the wavelength X = 2nc/u, and absorption is neglected

iv =0), that this equation takes the form

13
X

2

b

„ 2

-i = i

-^- - a + j yy, <2-3o> ^ X -X ^ X -X

n on n on

where the optical refractive index is given by

t\ 2

= «/£ = 1 + X (2-31)

o

according to eq. (2-5). For historical reasons eq. (2-30) is

referred to as Sellmeier's -Formula; it is widely used to

empirically fit experimental dispersion data.

In view of eq. (2-29) and definition (2-5) the permittivity

function for the Lorentz dielectric model is given by

2

£<u) - U1 *1

, 2 "r .—]. <2-32>

n ( w —w ) +j tov on n

If the restoring force is eliminated, eqs. (2-29) and (2-32)

describe the electrical behavior of a lossy metal:

2 _ w

X(w) = - > . pn

r , (2-33)

L w (w— iv ) J n n

14
2

w _ «(w) = € 1-Z , P"

r\ (2-34)

n n

Furthermore, with v set to zero, these constitutive parameters

also pertain to the electrical properties o-f a cold,

coll isionless plasma. Expressions similar to eqs. (2-32) and

(2-33) were originally derived by Drude C123 for a metallic

conductor.

A more rigorous quantum mechanical treatment of the atomic

system, given by Kramers and Heisenberg C133, leads to an

expression for e (w) which is identical in form to eq. (2-32).

In the quantum mechanical case, «hw corresponds to the energy

difference between two electronic quantum states; the reciprocal

of v relates to the lifetime of a transition between these two

2 states; and w is proportional to the oscillator strength for

pn r- r s this transition. Absorption occurs through the excitation of a

resonant transition and the subsequent dissipation of the

excitation energy to the remainder of the system through lifetime limiting mechanisms. At frequencies far from resonance, the

electronic excitations (oscillators) exchange energy with the

radiation field with practically no absorption taking place.

This exchange does, however, lead to a modification of the

propagation velocity as manifested by the refractive index

tj(w)=V£' (w)/€ , where «' is the real part of «. o

15
2.3 The Debye Dielectric Model

The literature concerning dielectric polarization is

dominated by two classical theories: the Lorentz theory for

resonant dispersion, and the Debye theory C103 for relaxative

dispersion. The latter deals with dielectric materials

containing structural elements (atoms, molecules, ionic radicals,

lattice defects, etc., collectively referred to as molecules)

which behave as non-resonant rotators possessing permanent

electric dipole moments. These dipolar moments tend to align

with the direction of an externally produced field, in opposition

to the randomizing effects of thermally induced collisions. The

frequency of collision is assumed to be so much larger than the

rotational frequency that the molecules can be regarded as

remaining stationary between collisions. As a result, the

resistance to motion is dominated by viscous, rather than

inertial, forces.

The original Debye calculation for the relaxation of polar

dielectric liquids was based upon a consideration of weak

collisions and Brownian motion. A particularly simple derivation

of the Debye formula will be given here for the case of strong

collisions. In a strong collision the molecule has no

recollection about its orientation or state prior to the

collision. The collision is also assumed to be adiabatic, i.e.,

to occur in a time frame much shorter than the oscillation period

of the impressed field. If the collisions are both strong and

adiabatic, the molecular orientations will obey a Boltzmann

16
probability distribution functionally dependent upon the

orientational energy of a molecular dipole within the impressed

field.

Consider a dipolar molecule belonging to the class of

molecules whose last collision occurred at time t when the

o

-> -» oscillating local field had magnitude E' (t ) = E'coswt . If the

o o

angle between the dipolar moment p and the field E' (t ) is 9,

then the potential energy of this molecule is given by:

-p-E' (t ) = -pE'coswt cos9. (2-35)

o

r o

The mean polarization per molecule, obtained as an average over

all dipolar orientations at the instant t , is directed along E'

and its magnitude is determined by the normalized Boltzmann

distribution:

I pcosQ exp (xcos9)sin9 d6

<p> = JO

exp (xcos9) sin© d6

f+1

p I , Y exp(xT)dy

« exp(xY)dy

(2-36)

The denominator is easily evaluated and, apart from the constant

factor p, the numerator is just the partial derivative of the

denominator with respect to

17
x = (pE'coswt )/kT. (2-37) o

Evaluation o-f the integrals leads to

<p> = p|cothx -

^-j, (2-38)

which for small x reduces to

2 2 +jwt

<p> = 3kf" coswt = %ia- Re e °- (2"39)

The mean polarization per unit volume, P(t), is obtained by

multiplying eq. (2-39) by the number of molecules per unit of

volume, N, and averaging over the time of the last collision.

With random collisions, the probability that the last collision

occurred in the interval Et-(0+dtf), t-#3 is (1/T>exp (-$/T)dtf,

where t is the mean interval between collisions. Consequently,

since P(t) is parallel to E'

,

2"* ^/4-% Np~-E' IfTM .. .. -0/T - P(t) = \L. ^ — cosw(t-*)e 3kT tJo

. N|!|l Re[e+J Mt^ e

" ( 1+J wt > * /Td«/T

]

Kl 2 + +jfa»t n ^Re E

!, . (2-40) 3kT L 1+jwt J

-* ~+ +iwt Writing P(t)=RePe J

, equation (2-39) yields

18
j. . Np2/3kT g,

1+jWT

-+ To express the local field E' at a dipole in terms of the

macroscopic applied field E, the generalized Lorentz-Lorenz

relation

E' = E + KP/«= (2-22)

o

is again introduced. Substituting this expression into eq.

(2—41) and solving for P now yields

P(w) = £ X(w)E(w), (2-42)

o '

where for the Oebye dielectric medium the dynamic susceptibility

function is given by

X(Q)

1+jWT X(w) = 7T , , (2-43)

and where

2 2

,N . r „ ,N . ,-1

*«» - HsM [» - H&t)] •

19
The Lorentz-Lorenz field correction there-fore has the effect of

enhancing the Debye static susceptibility constant X (0) and of

broadening the frequency response of the dielectric medium by

increasing the effective value of t. For K=0 the consequences of

this correction vanish.

In the case where the dielectric medium is composed of

several species of polar molecules, eq. (2-41) is replaced by

jt = (I ^!_)|^, (2-46)

n

J n

and eqs. (2-44) and (2-45) are generalized to read:

X<0) SkM1 NnPn

2

) V - SkM1 NnPn

2

)] ' (2"47)

on on

and

-1

T = T V o n - SET^P V„2

)]" • <2"48)

Eq. (2-43) remains unchanged except for these modifications.

A quantum mechanical derivation of the Debye dielectric

susceptibility function provides the same result. The quantum

20
mechanical derivation differs from the classical version only in

that the permissible dipolar orientations are discrete rather

than continuous. The average is now obtained by summing over the

various quantum states rather than integrating over a sphere as

in eq. (2-27). Either way,

£<W> = € fl+X(w)l = £ fl + ~4^-A. (2-49) oL J oL 1+jwt'J

21
3. PLANE WAVE PROPAGATION AND REFLECTION

The set o-f four Maxwell equations governing the

electromagnetic field arel

~* d B VxE = -—

,

(3-1)

-* 3D -*

7-B = 0,

(3"3)

7-D = p.

(3-4)

For a linear medium containing oscillatory sources and fields

these expressions simplify considerably. With the aid of

constitutive relations (2-5), (2-7), (2-8), and (2-9), and for a

sinusoidal field dependence such as given by eq. (2-13), Maxwell's

equations reduce to:

VxE(w) = -jwmH(m), (3-5)

VxH(w) = (<r+jw€)E(w) = jw£ +fE(w), (3-6)

22
V-tfH(w) = O, (3-7)

V- eE(w) = p (w) .

If the medium is also homogeneous, and if the net -free charge is

everywhere zero (p=0) , then the last two equations become

-» (3-9)

V-H = 0,

r, 2 ^ (3-10) 7-E = 0.

Under the stated conditions, viz., harmonic oscillation and

homogeneous, linear, charge-free space, the last pair of

equations is redundant with the contents of the first two Maxwell

equations. This is a consequence of a general theorem which

states that the divergence of any vector, which is itself the

curl of another vector, is identically zero. Accordingly, the

divergence of eq. (3-5) or eq. (3-6) leads, respectively, to eq.

(3-9) or eq. (3-10). On the other hand, the curl of eq. (3-5),

upon substitution of eq. (3-6) for VxH(w), gives:

Vx[vxE(w)l-w2

M€ eff E(w) = 0. (3-11)

To investigate the properties of plane wave propagation,

solutions to eq. (3-11) having the form

23
E(w) = E e

J (3-12)

w

are considered, where the vector constant E represents the

w

amplitude and direction of the oscillating -Field intensity at

-frequency w. When this expression is combined with the time

dependence o-f eq. (2-13) the complete space-time structure of the

solution becomes evident; namely,

jtwt- E(t) = Re E e

J

. (3-13)

-+ When k (w) is real, such solutions propagate with velocity w/k in

-» the direction of k while maintaining a constant peak amplitude

-» given by the magnitude of E . Furthermore, since the phase of

-» -+ E(t) remains constant over a plane perpendicular to k, this wave

is planar. The dispersion relation for plane wave propagation is

obtained by inserting eq. (3-12) into eq. (3-11):

kx(kxE ) + w

2

ji€ ^E = 0. (3-14)

w ef f u

Using a vector identity, this relation can also be written

as:

(k-E )k + (co

2

M€ ^ -k2

)E = 0. (3-15)

w ef f hi

For any given wave polarization and propagation direction this

-» dispersion relation provides the magnitude of k and its

dependence on frequency. From eqs. (3-10) and (3-12), or from

24
-» the scalar product of k with eq. <3-14) , it is evident that

k-E = (3-16)

u

The plane wave is consequently always polarized transverse to the

direction o-f its propagation, and eq. (3-15) reduces to

(w

2

M« ,,-k 2

)E = 0. (3-17)

ef f w

Since the presence of radiation implies E is not zero, the

w

quantity within the bracket must vanish. Evidently, plane wave

propagation is supported only if the fields are transversely

polarized and obey the law of dispersion:

k(w) = ±wVu~T~77wT. (3-18)

o eff

The magnetic field intensity associated with the plane wave

is readily determined by applying Maxwell's equation (3-5) to the

expression (3-12)

:

-+ -> - - -» u/\ r> -j k ( w )

• r k :£ -j k < w ) • r / -* « e, % H(w) = H e

J = x E e

J

. (3-19)

to mm u

o

Therefore, at each location within the radiation, the ratio of

the magnetic field intensity to the electric field intensity

remains constant at the value k/uji = V£ r .(w)/u . Furthermore,

o eff o

the directions of the electric field, the magnetic field, and the

propagation are always mutually orthogonal.

For plane wave propagation within an infinite isotropic and

homogeneous space all transverse directions of wave polarization

25
are equivalent. The presence of a surface will, however, break

the symmetry of the space and eliminate the degeneracy present in

the wave polarization. A preferred "plane of incidence" is

defined by the unit vector, n, normal to the surface and the

propagation vector, k. Relative to this plane an arbitarily

polarized plane wave can be resolved into two fundamentally

different, orthogonally polarized, plane wave components (figure

1). The plane wave component polarized perpendicular to the

plane of incidence is said to be s-polarized, or transverse

electric (TE) , and has as its field components:

E . = (kxn)-E . (3-20)

H , = V<s , _/m [kx (kxn)] -E (3-21)

til eff o fa)

where k=k/k is the unit vector in the direction of propagation.

The other plane wave component, polarized parallel to the plane

of incidence, is said to be p-polarized, or transverse magnetic

(TM) , and has as its field components

E „ = Lkx(kxn)J-E (3-22)

fa) 11 fa)

H „ = -7« ,,/m (kxn)-E . (3-23)

fa) It eff o fa)

At a planar boundary an incident wave of either polarization

generates both reflected and transmitted waves of the same

polarization. The amplitudes and directions of the reflected and

26
(vacuum) (medium)

(vacuum) (medium)

Figure 1. Reflection of a planewave at a material half -space,

(a) Field vectors for the transverse electric (TE) polarized

component. (b) Field vectors for the transverse magnetic (TM)

polarized component.

27
transmitted plane wave -fields are determined not only by the

incident wave, but also by the material parameters at the

boundary. In general, the tangential components of the total

electric and magnetic fields must each be continuous across a

surface. Since the s—polarized plane wave has only a tangential

electric field, and the p—polarized plane wave has only a

tangential magnetic field, the two forms satisfy different

boundary conditions and are reflected independently from the

surface. A different law of reflection applies in each case.

For the TE—polarized plane wave the (Fresnel) reflection

coefficient for the electric field at a vacuum-dielectric

boundary is given by £163:

E

r

. l-7l+X(w)/cos2

e. R (w) = -~ = . u

'. (3-24) S E

1

, l+Vl+X(w)/cos 8. w± 1

The superscripts "i " and "r" , respectively, refer to the incident

and reflected wave components of the total field, and Q. is the

angle of incidence defined in figure 1. For the corresponding

TM-polarized plane wave the (Fresnel) reflection coefficient for

the magnetic field at this boundary is:

H

r

„ Cl+X(w) ]-7l+X(w)/cos2

e. R (w) = -^ =

f

, i

*

. (3-25)

P H

1

,, [l+X(u)]+Vi+X(M)/cos e.

28
At normal incidence, when the angle of incidence 0. is zero and

R = -R . As expected, both re-flection coefficients reduce to

s p

the same expression because a plane of incidence can no longer be

defined. The minus sign is merely a consequence of the

conventions adopted in figure 1 for the positive field

directions.

29
4. INTEGRAL TRANSFORMATIONS AND THE FRESNEL THEORY

Classical Fresnel theory treats the reflection and

transmission of plane waves at flat, non—dispersive surfaces.

The generalization of this theory to include non—monochromatic

waveforms and dispersive media is readily accomplished by

introducing Fourier analysis methods. By means of the Fourier

transformation integral

-*i C m-+i -iwt

E (w) =

J E (t)e J dt, (4-1)

and its inverse

E (t) = — E (w)e J dw, (4-2)

an incident plane wave, having arbitarily specified waveform

-iE (t) at the reflecting surface is resolved into a spectrum of

truly monochromatic plane waves, each of (real) frequency w and

spectral amplitude E (to). Furthermore, if the reflecting medium

is assumed to respond linearly, each spectral component may be

treated independently using the Fresnel reflection coefficient

R(w) appropriate to the frequency and polarization of that

component:

-»r -*i (4-3) E (w) = R(w)E (w).

The reflection of the actual incident plane wave is subsequently

determined by superimposing the individually reflected spectral

30
components via the inversion integral:

E (t) = ;r— E (w)e J dw = ^— R(w)E (w)e d(0 . (4-4)

For the transverse-electric polarized plane wave the Fresnel

re-flection coefficient is given by eq. (3-24). A completely

analogous expression in terms of the magnetic -field can be

written for the transverse-magnetic polarized plane wave, with

R(w) given by eq. (3—25).

A further generalization of the Fresnel theory is achieved

if, instead of the Fourier transform, the Laplace transform is

employed. The Laplace transform facilitates the investigation of

transient phenomena initiated by the sudden arrival (at time t=0)

of a wave at the reflecting surface. The Laplace transform of

E (t) is defined by the pair of integrals:

->i f ~*i -st E (s) = E (t)e dt (4-5)

<„

s

'

and

->i 1 f T+j " E (t) = ^—- E (s)e ds. (4-6)

t E (s)e ds. 2nj J

Transform variable s=ff+jw is generally complex as variables <r

and to are both real. The number y , appearing as part of the

integration limits of eq. (4-6), is also real and is chosen so

that the integration path o=y in the plane of complex frequencies

31
+i

s lies to the right o-f all singularities o-f E (s) . The bounded

nature o-f any physical field at infinite times assures, however,

-»i that all singularities o-f E (s) occur in the left half-plane

where Re s$0. So, for all practical purposes, y approaches zero

as its minimum limit.

A useful interpretation of the Laplace transform is to

regard integral (4—6) as a superposition of exponentially growing

-+i and decaying sinusoidal components which combine to form E (t)

.

The amplitudes E (s) of these components are resolved using

integral (4-5) . The Fourier transform and the Laplace transform

Are very closely related. In fact, a formal conversion of

Fourier transform pair (4-1) and (4-2) into Laplace transform

pair (4-5) and (4-6) is readily achieved by substituting s for jw

-*i and recognizing that E (t) is zero until the incident wave

arrives at the surface at time t=0. This same substitution

analytically continues R(w) into the reflection coefficient R(s)

-+i for the spectral component E (s) . By analogy to eq. (4—4) the

entire reflected wave at the surface is given by

E (t) = -— E (s)e ds = -— R(s)E (s)e ds.

2ttj J ^.nj J t-j» t-jm (4-7)

The principle of causality prevents the reflection E (t) from

initiating before the arrival of the incident wave at t=0. This

observation may be applied to eq. (4-7) to demonstrate that

neither E (s) nor R(s) can have any singularities in the right

half of the complex s plane. The reflection coefficient for

32
treating TE plane wave transients is given by the analytical

continuation of eq. (3—24),

l-Vl+X(s)/cos2e. R(s) =

»

, (4-8)

lWl+X(s)/cos e.

where the dielectric susceptibility X(s) is some rational or

otherwise well—behaved function of s having no singular points

in the right half plane.

The inverse Laplace transform of R(s), given by the timedependent function

1 f Y+j " +st R(t) = ==-r

J R(s)e ds, (4-9)

has a simple and useful physical interpretation. Consider an

incident plane wave which has as its waveform a Dirac delta

"function" profile of unit amplitude:

E

X (t,r) = S (t - -cosO. - -sine.). (4-10)

c 1 c 1

Suppose the path of integration for eq. (4-7) is closed with an

arc of infinite radius surrounding the right half plane of s

where or is positive. For t<0, integration along this arc

contributes nothing to integral (4-7) because the integrand

exponentially vanishes here. On the other hand, the^Cauchy

residue theorem, together with the causal condition E (t=0) j£or

t<0, guarantee that no singularities of either E (s) or R(s)E (s)

occur within the enclosing contour of integration, i.e., the

right half plane. Furthermore, because E (s) has no

singularities in this portion of the complex frequency plane then

neither has R(s).

33
At some point z=0 on the surface x=0 the Laplace transform of

this waveform is

*<*> 1 (s) = f 5(t)e Stdt = 1. (4-11)

According to eq. (4-7) the amplitude of the reflected waveform at

this location is

T+j - +.t

E^>=^J> (t) = 7^-r R(s)e ds = I (t). (4-12)

Thus R(t) describes the temporal response of the surface to an

impulsive waveform of unit amplitude and is named the img.ul.se

C§s£QQse function.

For observation points off the surface the reflected wave

must display spatial as well as temporal dependence.

Generalization of R(t) is accomplished by recognizing that upon

reflection the normal component of an incident wave's propagation

vector is reversed. Consequently, the reflection of plane wave

(4-10) is readily constructed from R(t) by substituting for t the

retarded time

Ltl = t + (xcos8. - zsin0.)/c. (4-13)

l l

The generalized impulse response function RCtD is referred to as

the Greenes function, or transient ergp,agatgr, f°r the TE plane

wave reflection problem.

34
To be worthy of its title RCt3 must completely characterize

the elementary reflection process at any given observation point

and at any particular instant of time. To demonstrate this

property of RCtD observe that according to eq. (4-7) , for any

incident plane wave described on the surface by transform E (s)

the reflected plane wave anywhere in front of the surface is

given by

±r -» -*r 1 r Y+J w -h +s[t3

E (t,r) = E [t] = =*-r R(s)E (s)e ds. (4-14) 2nj J Y-jw

Substitution of definition (4—5) for E (s) and the corresponding

integral for R(s) and rearrangement of the integrations permits

eq. (4—14) to be converted into the following convolution

integral

E (t,r) ] E (T)R(Et3-T)dT. (4-15)

By means of expression (4-15) the transient reflection of any TE

plane waveform specified by E (t) can be computed once RCtJ is in

hand.

Integral (4-15) can be viewed as a superposition describing

the total reflection caused by a train of impulsive plane waves.

Thus, if the amplitude of the incident waveform appearing on the

surface within the infinitesimal time span t=r to t=r+Ar is

viewed as a Dirac delta function of strength E (t)at, then

-»i E (t) ATR(CtJ-T) is the amplitude of its reflection observed to

commence at location (x,z) at later time t = t- (xsin© . -zcosG .) /c.

35
Furthermore, the complete waveform E (t,r) is seen to

consist o-f a superposition o-f such elementary reflections from

all earlier times, i.e., from t=0 until the present,

E

r (t,r) = EX (0)ATRCt3 + ...

+ E

1 (r)ATR<Ctl-T) + ... + E

1 CtjATRCO:

m

n=0

-5*

Equation (4-15) is recovered in passing to the limit in which

m = Ctl/Ar -» co as At -» 0, while mAT = Ltl remains fixed.
5. CONSTRUCTION OF THE TRANSIENT PROPAGATOR

An algorithm will now be presented -for constructing the

transient propagator describing TE plane wave re-flections into

vacuum. The reflecting medium is assumed to be a non—magnetic

dielectric half—space modeled by a complex susceptibility

function, X(jw). If the reflecting medium has any conductivity

it is included in X(jw). Under these circumstances the

conventional Fresnel reflection coefficient, originally presented

as eq. (3-24), applies:

o, ^ - l~>/l+X<jM) R(jw) = >

'

' —> (5-1)

1+Vl+X(jw)

where

X(jw) = X(jw)/cos2

e.

(5-2)

To facilitate computations, the function R(jw) is

rationalized by multiplying both numerator and denominator by

1-vl+X ( jto) . This leads to the expression:

R(jw) = -1 + -^-2 f-lWl+X(jw)l. (5-3)

X(jw) L J

The radical remaining in expression (5-3) is eliminated by

expanding it as a binomial series:

37
(3/2) r ,n+l

R(jw) = ^ Y (3) " t-X(jw)] - (5-4)

n=0

Appearing within the summation is the Pochhammer symbol de-fined

by

(z) = r<z+n)/r<2) = 2(2+1) (z+2) ... (2+n-l). (5-5)

n

r(z) is the gamma -Function, which among its many properties, is

best known as the generalization of the -factorial function to

non-integer and generally complex numbers.

To address the transient problem, reflection coefficient

(5-4) is analytically continued into the complex frequency plane

by replacing jw by the Laplace transform variable s:

" (3/2) _ n+1

R(s) = i

) L-X(s)J . (5-6)

n=0

As eq. (5-6) stands, R(s) is expressed as a summation of rational

functions in the variable s. Term by term inversion of R(s) in

this form can now be readily accomplished. In certain cases it

will be possible to sum the resulting time series for R(t). In

other cases the impulse response function must be approximated by

considering an appropriate number of terms in the series.

Construction of the transient propagator is completed by

replacing the argument of R(t) by the retarded time Ct3.

38
According to eq. (4-13)

,

Ct3 = t + - cosO. - - sine.. (5-7)

C 1 C 1

Once RCtJ has been determined for a given material it may be

used in conjunction with convolution integral (4-15) to predict

the reflection of any specified, incident, plane waveform. The

remainder of this report is devoted to applying this algorithm to

the dispersive materials modeled in chapter 2.

39
A. THE TRANSIENT PROPAGATOR FOR TREATING REFLECTIONS FROM A

LORENTZ DIELECTRIC

The frequency-dependent susceptibility function for the

Lorentz dielectric model was presented in eq. (2-29) for a

material composed of several different kinds of oscillators. In

the present discussion the Lorentz dielectric will simply consist

of a single species of oscillators, viz., electrons bound to

their atomic cores with harmonic frequency w (binding energy

4

-ttto ). For this case eq. (2-29) reduces to

o

2

to X(jw) = 1 . (6-1)

(w —to )+jwv

o

J

As earlier, the strength of the collective interaction between

the oscillators and the electromagnetic field is represented by

the second power of the plasma frequency w . This quantity is

proportional to the average charge density of the oscillator

electrons. The oscillator motion is also subject to a viscous

damping force characterized by the collision frequency, v. The

latter manifests itself either as a linewidth in a spectral

measurement, or as a reciprocal lifetime in a relaxation

measurement

.

To construct the transient propagator RCtJ the dielectric

susceptibility function X(jto) is analytically continued from the

4 Due to corrections, such as given by eq. (2-28), accounting for

the presence of neighboring oscillators, w is not precisely the

binding frequency.

40
imaginary axis into the plane of complex frequencies by replacing

jw by s in eq. (6-1).

W to X(s) = 2A A 2

S +VS+W

o

(S-Sj ) (s-s_)

(6-2)

The poles of X(s) located at s and s_ are observed to be real

when v£2w , or else complex conjugates when v<2w :

5 1' 52

=i

-f*Vf^

"2 = "M

o

k v J 2 /vV

"2 * J>

M

o

"

(2)

, f or v >2w

s for v=2«

for v<2w (6-3)

The temporal response of the dielectric is determined by the

poles of its susceptibility function. For example, the inverse

Laplace transform associated with a simple pole is an exponential

having this pole as its characteristic frequency,

s t

L' l [7i?TT]=-

a

- (6-4)

Referring to eq. (2-15), the poles of the oscillator displacement

function r (s) are seen to be identical to those of X (s)

Therefore, the dielectrical behavior of the medium as v is varied

can be directly interpreted in terms of the oscillator motion.

41
The locus of the poles in the complex s plane, as v is

increased -from zero through in-finite values, is shown in -figure

2. For v=0 the poles are located on the imaginary axis at s=±jw

and the oscillator electrons vibrate with resonant frequency w . As v ranges from zero to v=2w the poles move as complex

conjugates in the left half plane along a semi -circular

trajectory of radius w centered at the origin. Within this

range the electronic motion is underdamped, i.e., it is

oscillatory but exponentially attenuated with increasing time.

For v=2w , the poles coalesce on the real axis at s = -v/1 = —w . o' r o

Now r(t) is critically damped, reflecting a transition to nonoscillatory motion. For w>2w the motion is heavily damped and

characterized by two real time constants; one approaching zero,

the other approaching infinity, as v+m. The pole locus therefore

defines three distinct regimes for investigating reflections from

simple Lorentz dielectrics. These regimes correspond to

dielectric media which are underdamped (v<2« >, critically damped

(v=2w ), and overdamped (v>2w ).

o

r o

To continue with the construction it is necessary to

identify X(s) in eq. (5-6). According to eq. (5-2),

- 2

X(s) = X(s)/cos 6. - r= , (6-5)

1 (S-Sj ) (s-s_) *

where w = u /cosd . . Therefore

p p l

42
I

Im s

s-plane

^

Res

P=

Figure 2. The locus traced by the poles of X(s) as the inelastic

collision frequency v is varied between zero and infinity. For

0$v$2oj , the roots follow a circular locus of radius w with

9=±costV/2m ). For

half of the real axis.

'.w the locus coincides with the negative

43
R(s) = -=<

- , "

(3/2) (w ) 2n « ) ) (-1) ~tt\ 4 Pp L <3)_ «3) , ,n+l ; x n-frl

" n=0 n <S-5

1

) (S-S2> (6-6)

Note that the simple poles of r (s) and X <s) appear as simple and

biQt!!?!! 9!lder BQL§:5 i° the power series expansion of R<s). As a

result R(t) will contain terms more complicated than the exponential

functions given by eq. (6-4). Nevertheless, the inversion of

R(s) is readily accomplished by applying the following Laplace

transform pair (A+S 29.3.50) term-by-term to eq. (6-6):

f(k) r-l t \k-l/2 -(a+b)t/2 ^VF(-M k - 1/2 e" <a+b>t/2

I fazbt\ . \a-b/ k-l/2\ 2 )

'

t ^ > k / ^i,% k Ck>0) (s+a) (s+b) (6-7)

The last factor is a modified Bessel function of the first kind,

of argument (a—b)t/2 and order k-1/2.

In the overdamped regime (v>2w ), the poles are real and,

o

according to eq. (6—3), given by

v I 2 2~~

s t ,s„ = —= ± a, where a = i (v/2) -w . (6-8)

The corresponding impulse response function is determined by

applying transformation (6-7) to expression (6—6). Since n is an

integer, T(n+1) = n! and

5 Parenthetical entries of this form refer to formulas tabulated

in the Handbook of Mathematical Functions, li. Abramowitz and I.

A. Stegun; c.f. References at conclusion of this report.

44
co —C

r^ mi. x « , t- (3/2) _(« t) n n . ,_ R(t) 1 - .,/ V , <,n n r P 1 n _ \ -vt/2 = --r(w t)< > (-1) t=t „

r . =—7- I L « ._(at) > e - 4 p \ Z (3) n! L 2at J 2at n+1/2 / W

p n=0 n (6-9)

The impulse response function consequently appears as a summation

o-f products each consisting of a numerical coefficient, a

monomial, a modified Bessel function, and an exponential factor.

By introducing the modified spherical Bessel function:

i (at) = J^r- I ^. ._(at) , (6-10)

n i2at n+1/2

eq. (6-9), for the overdamped dielectric (v>2w ), is simplified,

D/4-\ i / V- (3/2) _(w t) ., . .,_

P n=0 n (6-11)

Expression (6-11) is dimensionless because, as in eq. (4-15),

R(t) has the dimensions of reciprocal time.

In the underdamped regime (v<2w ) , the poles are complex

conjugates and, according to eq. (6-3), given by

v I 2~~ 2

s ,

= -— ± jw , where w s Vw -(v/2) = ja . (6-12)

Comparison of eq. (6-8) with eq. (6-12) shows that result (6-11)

can be adapted to the underdamped dielectric case by replacing a by -jw . Since

c

45
i (at) = i <-jw t) = <-j) n

j (u t) , (6-13)

n n c n c

where jJ (w t) is a spherical Bessel function of the first kind,

n c r ' the impulse response function for the underdamped dielectric

(v<2w ) can be written as

o

w — 2

o/j-x « # *- (3/2) _(w t) _n . . ._ R(t) 1 - . . / V , «.n nf p 1 . , . .\ -vt/2 = —-(w t)< ) (-1) y=r rl _r . j (w t) > e - 4 p \ Z (3) n'L 2« t J J n c /

p n=0 (6-14)

Expressions (6-11) and (6-14) describing the reflection of

an impulsive field are reminscient of a modal expansion; the

higher order modes corresponding to higher orders of the simple

poles appearing in r (s) and X (s) . Furthermore, the appearance of

i (at) in (6-11), or j (w t) in (6-14), is readily associated

n

J n c' with, respectively, the monotonic decay in the motion of an

overdamped oscillator, or the vibratory decay of an underdamped

osci 1 lator

In the critically damped regime (v=2w ), the Lorentz

dielectric medium is characterized by a real, second order pole

at s = -v/2 = -to , corresponding to a = w =0. For small

arguments, the limiting value of the spherical Bessel function

(A+S 10.1.4, 6.1.12, and 6.1.22) is given by

46
( W t) n

j (w t) -+ (6-15)

n C 2n (3/2)

n

With this substitution, eq. (6-14) reduces to

^ ,w t. . 1 . r ,w t v 2.

-H-f-H i BF^i-m n

^t/2

• w - n

p n=0

The quantity within the curly brackets is recognized as being the

2 defining series -for the hypergeometric -function _F C3;-(w't/2) 3,

so that

R(t) -'"-*' M t2

P

^.['--(Hr2

-

Use of the identity (A+S 9.1.69):

V*> =F<^T7(l)ViH'-(l)

2

]

permits eq. (6-17) to be re-expressed in terms of Bessel function

J n (u't). 2 p

R(t) — =

_ —-2 _ - . , -vt/2 ,, 10 , J (w t)e . (6-19)

w (w t) P

P P

47
Apparently, the coalescence of the two simple pol^s o-f r (s) and

X (s) into a single second order pole in the critically damped

case coincides with a corresponding condensation of the modal

expansion into a single oscillatory term.

Thp final step in constructing a transient propagator RCt3

requires the substitution of eq. (5-7) for t in the impulse

response functions R(t). Namely, the substitution of

CtU = t + (xcose -zsin9 )/c (6-20)

1 i

for t in eqs. (6-11), (6-14), and (6-19). The behavior of these

transient propagators is illustrated as a function of the

retarded time in figure 3 for a choice of parameters. At the

point (x=0,z=0) on the dielectric surface, the reflected pulse is

first observed at time t=Ct3=0. However, at a location situated

in front of the dielectric surface, where x is negative, the

reflected pulse does not appear until Ct3=0, i.e., not until time

t = (Ix lease. + zsinO.)/c.

i l

48
0.20

a 0.10 h

'3

-0.10

-0.02

DC

1 1 < 1 1 T

-

/ (a)

a> = 0.89443 COp

—

,.,_,. 1,

V = 0.12522 a> p

i i i i i -

DC

i^ i 1 i

1 1

-0.02 /

(b)

/ (x) = 0.89443 aj p

-0.04 / V =2C0 = 1.78886aJ

p_

-0.06

-0.08

1 i i i i i i

cl -0.02

'3

lTj -0.04

DC -0.06

0.08

(c)

a> = 0.89443 aj p

y = 3G> = 2.68329 a)

p

± ± ±

10 15 20

p[t]

±

25 30 35

Figure 3. The transient propagator for treating re-Flections -from

a Lorentz dielectric hal-f-space. (a) Under damped medium, (b)

Critically damped medium, (c) Dverdamped medium. The parameters

in case (a) were chosen to agree with those used by Sommer-feld

and Brillouin C2D

.

49
7. THE TRANSIENT PROPAGATOR FOR TREATING REFLECTIONS FROM A

DRUDE METAL AND A PLASMA

The Drude model of a metal describes electrical conduction

in terms of the motion o-f an electron gas which undergoes

momentum dissipating collisions with an ion lattice it freely

permeates. The lattice functions to provide charge neutrality to

the metal. Band structure, anisotropy, spatial dispersion, and

other properties of the solid state are totally ignored. As a

consequence, the mathematical description of a Drude metal is

identical with that of a gaseous plasma to be found in the

ionosphere, the sun, or a thermonuclear fusion device.

The properties of a Drude medium are readily deduced

from the Lorentz model by setting the binding frequency w of the

dielectric oscillator electrons to zero. The susceptibility

function which results from performing this operation is given by

eq. (2-33), and for a single species of electrons is

2

—to X(jw) = —r-2

r . (7-1) w(w—JV)

The reduced susceptibility function appearing in eqs. (5—1) and

(5-2), when expressed in terms of the complex frequency s, viz.,

_ 2

w X(s) = A , , <7-2) s(s+v)

has real poles at s=0 and s=v, with w =w /cosG . . While one pole

p p l remains fixed at the origin, the other travels with increasing v along the negative real axis from the origin to negative

50
infinity. By having set w to zero the semi -circular portion of

the pole locus, shown in -figure 2 -for a Lorentz dielectric, has

been collapsed to a point at the origin. The remaining portion

of the pole locus, ascribed earlier to an oyerdamped Lorentz

dielectric, now corresponds to the Drude metal. The impulse

response function for the Drude metal is therefore most easily

determined by placing w =0 in expression (6-11). In that case,

according to eq. (6-8), a=v/2 and

<» - 2

^,*_. , ^ (3/2) r (w t) ..n . , ._ R(t) 1 - ../ V , «,n nr P I , . , ,\ -vt/2

P n=0 ° (7-3)

As already mentioned, the description of a cold,

coll isionless plasma as a tenuous electrically neutral gas of

electrons and positively charged ions is identical to the Drude

model of a metal when v=0 and the motion of the much more massive

positive ions can be ignored. For v=0, both poles of X (s) are

fixed at the origin of the complex frequency plane. The impulse

response function for the plasma is determined by letting v*0 in

eq. (7-3). In this limit the exponential approaches unity and

the Bessel function approaches (A+S 10.2.5, 6.1.12, and 6.1.22)

(vt/2) n

i (vt/2) -» . (7-4)

2

n (3/2)

n

As a result, eq. (7-3) reduces to

51
R(t) = l/"p*]

"

w' 2

P n=0

, ,M t, r -w t. 2..

By means of identity (6-18) the hypergeometric -Function appearing

in expression (7-5) can be replaced by a second order Bessel

function, with the result that for the lossless medium

^P- = —2 j^(w t). (7-6)

w (w t) P

P P

This particular impulse response function, derived by other

means, has been used to interpret transient returns from

ionospheric sounding experiments C83.

The transient propagators for the Drude metal and plasma are

obtained from eqs. (7—3) and (7—6) by replacing t with the

retarded time function Ct3 defined in eq. (5-7), as

CtJ = t + (xcose. - zsine.)/c. (7-7)

l l

A normalized graph of the transient propagator versus retarded

time appears in figure 4 for the Drude metal (v=0. 1258w ). The

52
-0.30 -

CO, [t]

Figure 4. The transient propagator for treating re-flections -from a Drude metal half -space. The considered value of v/w

corresponds to the underdamped Lorentz dielectric propagator

shown in figure 3(a) with u =0.
transient propagator for the lossless plasma <v=0) has a very

similar appearance except that the oscillations are slightly

larger and less damped in this case.

54
8. THE TRANSIENT PROPAGATOR FOR TREATING REFLECTIONS FROM A

DEBYE POLAR DIELECTRIC

The Debye polar dielectric material, typically an organic

gas or liquid, is modeled as an ensemble of permanent dipolar

molecular elements attributed with rotational freedom. These

elements tend to rotate into alignment with an applied field,

against opposing thermal forces which tend to randomize the

molecular orientations. In comparison with a Lorentz dielectric,

a Drude metal, or a neutral plasma, relaxation processes are

generally slower in the Debye dielectric because the molecular

mass of the rotator greatly exceeds the electron mass of the

oscillator and because the thermal restoring forces are generally

weaker than those of electrostatic origin.

According to eq. (2-43) the Debye dielectric susceptabi 1 i ty

function is given by

X(jw) =

* (0>

, (8-1) 1+jWT'

where X(0) is the static susceptibility constant given by eq. (2-

44) and r* is related by means of eq. (2-45) to the mean time

interval between collisions. For economy of notation the prime

appearing on r will hereafter be suppressed; the appearance of t

will signify the value of t' as defined by eq. (2-45).

In order to construct the transient propagator RCtD it is

first necessary to replace jw by s in (8-1):

*<*> = Ti§T777 •

and then to define

55
X (s) = (8-3)

T(S+1/t)

where

t = tcos e. /X(0)

1 (8-4)

Clearly the character of the pole at s= -1/t is not influenced by

the magnitude 3 o-f 9. or X (0) . l In terms of definition (8-3) the transient reflection

coefficient determined by eq. (5-4) for susceptibility function

(8-3) is

(3/2)

R(s i X

n [ -1 ]n+l

4/(3) - . . . _ n Lt (s+1/t) J n=0

(8-5)

Inverting this expression, term by term, by means of the Laplace

transform pair (A+S 29.3.11):

T<k) k-1 -at

(s+a) k k>0 -t e (8-6)

immediately leads to the impulse response function:

(3/2)

««» = £ i 1 ST^") •"*" (8-7)

n=0

The series contained within the curly brackets is simply the

56
confluent hypergeometric function and so

R(t) = -3 .F <3/2; 3; -t/T)e t/T

. (8-8)

4t

The confluent hypergeometric function F (a;b;z) appearing in eq,

1 1 (8-8) is identical to Kummer's function li(a,b,z). In the

notation of Abramowitz and Stegun (A+S 13.1.2 and below 12.1.10):

F (3/2; 3; -t/r) 3 M(3/2, 3, -t/r) . (8-9)

Furthermore, it is extremely useful to note that the Kumrner

function appearing in (8-9) satisfies, with v=l and z=—t/2T the

identity (A+S 13.6.3):

M(i/+l/2, 2v+l, 2z) = T(l+v) (2/z) W

I (z)eZ. (8-10)

v

Thus

M(3/2, 3, -t/T) = -4

J- I (~le t/2T

, (8-11)

V 2t'

where I (-t/2r) is the modified Bessel function. Substitution

of eqs. (8-9) and (8-11) into eq. (8-8) now yields

R(t> -

+

E

I i@)""*[-(7 + ^H • <e- 12 '

Since I (-z) = —I (z), the impulse response function for the

Debye dielectric assumes as its final form:

57
TR(t) „ _ Lj^y^pf^I + i_) t]. (8. 13)

The transient propagator is obtained by replacing t in this

expression by relation (5-7) for CtD. The behavior of RCt3 is

graphed in figure 5 for several values of t/t.

58
-1 -

-2

DC

-3

-4

-5

1.25

Figure 5. The transient propagator -for treating re-flections -from

a Debye dielectric hal-f-space. Four cases are shown, each

parametrized by a di-f-ferent value o-f the static susceptabi 1 i ty

-function X (0) divided by the squared cosine function of the

incidence angle 9..
9. SUMMARY AND CONCLUSION

The transient propagator (or Green's function) dealing with

the reflection of an incident, impulsive, plane wave is the

fundamental solution to the time-dependent plane wave reflection

problem. By means of this function the reflection of an incident

plane wave of arbitarily specified waveform can be calculated via

a convolution integral. Construction of the transient propagator

for a transverse—electric polarized plane wave was implemented by

an algorithm devised to facilitate inversion of the Laplace

transformed impulse response function. This algorithm was

applied to study reflections from four frequency—dispersive model

material surfaces; the Lorentz dielectric; the Debye dielectric;

the Drude metal; and the cold, col 1 isionless plasma. Table 1 lists the impulsive response functions which were obtained. The

transient propagators, determined by replacing t in these

expressions by the retarded time Ct3, are shown in figures 3, 4,

and 5. Corresponding expressions for transverse-magnetic

polarized fields, or other material surfaces may be derived using

similar techniques.

60
L

01 4J

ItE

01>

•H

inL

01a

Ul •Hai>.uc

at3

cr

aiL

inc

•H

-Muc3

01

inca

inaiL

at

i/i i-t3 in a ai £ u

•H *

01

L

r 3

h- in

CN

4J

JL.

CNI

4J zoUzD

U.

LU

Ulzo

Q.wW

cc

Ul_JD

CLE

CNI 3

CN

v.

CN 3M

CNSI I 3

CN

v.

a

a

CN

K) I r ,0 8 I^VJ IIC

13

-It I

CN

CNI

II

•P I o 8 U"V] IIC

13I

II I 3 P I 3 EDa

UJE LPU

01

l-H01 •HQN

+Jc

QJL

_J

"O

01aE

TJ T3L

01 "OcD

01aEaL

CJ

XI

01aE*nL

01>o

61
JL.

cn

13ca iM 8 [X^J

13

-It I

II

4J

at E

•o3LD

CS

CN

rs I 3 6

in

ui

atCu

fsi

— I k- a

QJ I k- CN

k-l-P I

II

k-U

•wL

4Ju

0)

01 QL

<U

t-t

LL

01

X)

01 Q

N

IN 3 X 1 1 v. 4

'1

CN cn

•v. in i

*^

u

kII ii H It-N

r-> v. (D >

in

^•^1 u CM

X.

^i a. .3 3

II II

13

CDZo

l-t I-z

LLLUa

N

62
10. ACKNOWLGEMENT

The author wishes to thank J. F. Marchiando -for performing

the numerical computations leading to -figures 3, 4, and 5 and for

providing many helpful suggestions throughout this project.

63
1 1 . REFERENCES

1. Fork, R. L. ; Shank, C. V.; Yen, R. ; Hirlimann, C. A.

Femtosecond optical pulses. IEEE Journal of Quantum

Electronics QE- 1 ? < 14) : 500-505; 1983.

2. Brillouin, L. Wave propagation and group velocity. New

York: Academic Press; 1960. 154 p.

3. Wait, J. R. ; Froese, C. Re-Flection of a transient

electromagnetic wave at a conducting surface. Journal of

Geophysical Research 60(1): 97-103; 1955.

4. Dudley, D. G. ; Papazoglou, T. M. ; White, R. C. On the

interaction of a transient electromagnetic plane wave and a

lossy half-space. Journal of Applied Physics 45(3): 1171—

1175; 1974.

5. Chen, S. N. C. The transient phenomenon in an isotropic

plasma without collision loss. Proceedings of the IEEE

51(7): 1045-1046; 1963.

6. Knop, C. M. Comments on "The transient phenomenon in an

isotropic plasma without collision loss." Proceedings of

the IEEE 52(1): 99; 1964.

7. Knop, C. M. Further comments on "The transient phenomenon

in an isotropic plasma without collision loss." Proceedings

of the IEEE 53(7): 751-752; 1965.

8. Wait, J. R. Oblique reflection of a plane impulsive

electromagnetic wave from a plasma half -space. Physics of

Fluids 12(7): 1521-1522; 1969.

9. Price, G. H. Reflection at oblique incidence of an

impulsive transverse magnetic plane wave from a cold, lossy,

semi-infinite plasma. Radio Science 8(12): 1165-1168; 1973.

64
10. Debye, P. Zur theorie der anomalen Dispersion in Gebiete

der langwelligen elektrischen Strahlung. Verhandlungen der

Deutsche Physi kal ischen Gesellschaft 15(16): 777-793; 1913.

11. Larentz, H. A. The theory of electrons and their

applications to the phenomena of light and radiant heat.

2nd ed. New York: 6. E. Stechert and Co.; 1916. Chapt. IV,

132-167; Lecture note 57, 309-311.

12. Drude, P. Zur Ionen—Theori e der Metal le. Physi kal ische

Zeitschrift 1(14): 161-165; 1900.

13. Kramers, H. A.; Heisenberg, W. Ueber die Streuung von

Strahlung durch Atome. Zeitschrift fuer Physi k 31(9): 681-

708; 1925.

14. Lorentz , H. A. Ueber die Beziehung zwischen der

Fortpf lanzungsgeschwindigkei t des Lichtes und der

Koerpersdichte. Weidemann, Annalen der Physi k 9(4): 641-665;

1880.

15. Lorenz , L. Ueber die Ref ractionsconstante. Uieidemann,

Annalen der Physi k 11(9): 70-103; 1880.

16. Born, M. ; Wolf, E. Principles of Optics, 6th ed. New York:

Pergamon Press; 1980. 36-51.

65

NBS-114A (rev. 2-eo

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS/TN-1202

2. Performing Organ. Report No 3. Publication Date

March 1985

4. TITLE AND SUBTITLE

Transient Analysis of Electromagnetic Reflection From Dispersive Materials

5. AUTHOR(S)

A. George Lieberman

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions]

National Bureau of Standards

Department of Commerce

Gaithersburg, MD 20899

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Same as Item 6

10. SUPPLEMENTARY NOTES

3 Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

Theoretical expressions are presented describing the transient and steady-state

temporal evolution of an impulsive, TE-polarized, plane wave reflected into vacuum

from any of a variety of frequency-dispersive material surfaces. Polar dielectrics,

non-polar dielectrics, metals and plasmas, are treated using, respectively, the

Debye, Lorentz, and Drude material models to investigate the effects of dispersion

upon dimensional measurements performed with optical pulses of extremely short

duration. The more general problem, concerning the reflection of an optical pulse

of any specified waveform, is resolved by convoluting the incident wave at the

material surface with the previously determined reflection of an impulsive wave.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Debye dielectric; dispersion; Drude metal; electromagnetic fields; Green's function;

impulse response; Lorentz dielectric; material models; optical pulses; plasma;

reflected waves; transient fields; waveform analysis

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