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Special Report 10

RADAR REFLECTION COEFFICIENTS FROM A PLASMA GRADIENT WITH COLLISIONS (U)

fly; WALTER G. CH1 SNUT

Prepared for:

DEFENSE ATOMIC SUPPORT AGENCY WASHINGTON, D.C. 20305

r -^ : F£e 241959 '/

a t X

CONTRACT DA-49-146-XZ-184

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NOTICE

This document has been approved for public release

and sale; its distribution is unlimited.

I

lb *II ■ .■ *- - - ■ J

DASA 2185 October 1968

Special Report 10

RADAR REFLECTION COEFFICIENTS FROM A PLASMA GRADIENT WITH COLLISIONS (U)

Prepared for:

DEFENSE ATOMIC SUPPORT AGENCY WASHINGTON, D.C. 20305

CONTRACT DA-49-146-XZ-184

By: WALTER G. CHESNUT

SRI Project 4417

Approved: RAY L. LEADABRAND, MANAGER Radio Physics Laboratory

D. R. SCHEUCH, VICE PRESIDENT Engineering

Copy No.

ABSTRACT

This report presents numerical results in the form of graphs of the

power reflection coefficient for electromagnetic signals normally inci-

dent upon a plasma gradient. An electron collision frequency that is

independent of position in the plasma interface is assumed. A "slide

rule" Is provided to enable quick determination of the magnitude of the

reflection coefficient for "very overdense, exponential plasmas" in

the lower atmosphere (to 70 km) for frequencies from 30 MHz to 10,000

MHz. For plasmas that only reach a finite density, a complete set of

graphs is given presenting power-reflection coefficients for normalized -20 plasma parameters. Reflectivities are plotted if greater than 10

-3 +3 Normalized collision frequencies (Z) from 10 ' to 10 * are covered.

Normalized plasma to RF frequency ratios inside the plasma cover the -4 +4 range from 0.01 to 100 (10 ^ X ^ 10 ). Characteristic distances

over which the plasma density exponentiates In the gradient region vary -3 +3 from 10 wavelengths to 10 wavelengths. Simple aids to the reader

are provided so that he may easily determine his parameters for use

with the reflectivity curves, should his plasma be of atmospheric

constituents.

iii

■ - i --*-■ - -• i

CONTENTS

ABSTRACT

LIST OF ILLUSTRATIONS

LIST OF TABLES

ACKNOWLEDGMENTS

1 INTRODUCTION

2 BACKGROUND

3 REFLECTION FROM AN EXPONENTIAL PLASMA GRADIENT . . ,

4 REFLECTION FROM A GRADUAL GRADIENT TO FINITE ELECTRON DENSITY. ....

5 CONCLUSIONS

Appendix—REDUCTION OF ANALYTIC FORM FOR REFLECTION FROM A GRADIENT IN PLASMA DENSITY

REFERENCES

DISTRIBUTION LIST

DD Form 1473

iii

vii

vii

ix

1

5

15

19

77

79

89

91

i«* fc»

ILLUSTRATIONS

Flg. 1 Correcting Z to Zeff for use In Appleton-Hartree Formula—Procedure after Shkarofsky (Ref. 2) . , . . 6

Fig. 2 u)p/u) = /T—Corrected for use in Appleton-Hartree Formula—Procedure After Shkarofsky (Ref. 2) . . . . 7

Fig. 3 Electron-Neutral Colllsional Frequency vs. Altitude 9

Fig. 4 Electron-Neutral Collision Frequency Expressed in Megahertz 10

Fig. 5 Plasma Frequency vs. Electron Density 11

Fig. 6 Electron-Ion Collision Frequency vs. Electron Density 12

Fig. 7 Overlay for use with Fig. 8 for Calculation of Reflectivity from Exponential Plasma 16

Fig. 8 Slide Rule for Calculating Reflection Coefficient for Exponential Plasma Gradient 17

Fig. 9 Power-Reflection Coefficient vs. the Ratio oA for Values of Z from 10"3 to ID3 21

Fig. 10 Power Reflection Coefficient vs. Z for Values of oA from lO-3 to 103 47

Fig. 11 Comparison of Power Reflectivity vs. Z for a Gradual Gradient and an Exponential Profile .... 76

TABLES

Table A-l Test Cases to Evaluate Accuracy of Hybrid Analytic Approximation to Continued Product 87

vll

•*- --

1

t

I ACKNOWLEDGMENTS

(

The author specially appreciates the careful computer programming

provided by Mrs, Lorie McNiel. Mrs. McNlel was responsible for the '

formating of the computer drawn graphs so that optimum choices of plasma

frequencies would be used In order to avoid large gaps or dense crowding

on the graphs. She was also responsible for the "log-wise" stepping of

the abscissa values in order to obtain smoother—and thereby—more useful

plots.

Mr. Dennis Allison has checked the author's mathematical steps and

procedures for accuracy.

ix

., >. . .A

—■ r

1 INTRODUCTION

This report presents numerical results for the power-reflection

coefficient for electromagnetic waver normally incident upon an inter-

face between vacuum and a plasma. Specifically, numerical results are

presented for situations in which the plasma density varies gradually

over a wavelength of the electromagnetic wave. Collisions between the

plasma electrons and a neutral background gas are Included .

Two types of plasma density profiles are considered. In the first,

the plasma density is allowed to exponentiate to infinite density. For

many applications, this approximation is adequate because in practice

wave attenuation prevents the Incident signal from actually interacting

significantly with the very high electron densities existing at great

penetration depths. Therefore the details of electron density very deep

within the medium do not matter. The solution for the power reflectivity

in this case is a very simple analytic expression that admits to simple

evaluation. The power-re fleet ion coefficient in a gradient of this sort

is presented in the form of a "slide rule" to allow the reader to choose

his own exponentiating distance. The reflection coefficient is presented

as a function of altitude in the earth's atmosphere for frequencies from

30 MHz to 10,000 MHz.

The second density profile that is considered is one form of the

"Epstein" profile—a profile that permits, in principle, a very wide

latitude in choice of plasma layer shapes. In practice, a numerical

solution requires evaluation of rather unfamiliar analytic functions.

For our example, the function is the gamma function of complex argument.

The plasma profile considered here varies monotonlcally from vacuum to a

finite electron density. The electron collision frequency is assumed

to be constant throughout the plasma (electron-neutral collisions are

assumed to dominate) . For very high electron densities deep within the

plasma region, this particular profile appears to be nearly the same as

the exponential profile considered in the first part.

m fc

The determination of the absolute magnitude of gamma function of

complex arguments used In evaluating the reflectivity of thn Epstein

layer requires evaluation of a contInued product. A computer test has

revealed that for many cases of Interest In this work, 10,000 terms of

the continued product have not converged the reflectivity to within 20

percent of the proper result. We have therefore developed a technique

of analytic approximation to the continued product that allows very

rapid convergence to one-percent accuracy. Details are given in the

Appendix. The technique may be of interest to others faced with evalua-

tion of similar analytical functions. One form of our analytic technique

enables slide-rule calculation of power reflectivity to an accuracy of -20

10 percent should the reflectivity be greater than 10 . Therefore the

reader doer have at his disposal a rather simple, analytical technique

that he may apply for situations of particular Interest not covered by

our normalized graphs.

Section 2 of the report presents background information concerning

plasma properties. Section 2 also provides disclaimers, reservations,

admonitions, and qualifications reflecting the author's own practical

point of view regarding reflectivity calculations. Section 3 describes

the reflectivity from an exponential plasma gradient to infinite elec-

tron density. Section 4 presents the results of calculations for

gradients to finite electron density. The mathematical developments

are contained in the Appendix.

This report will show that coherent reflection from an interface

between a plasma and a vacuum can be extremely reduced if the interface

is gradual. The coherent reflection can become very small for some

plasmas that actually have very high electron densities. In this cir-

cumstance, it seems likely that energy can be returned to the s!.gnal

source by any one of many other kinds of reflection mechanisms. In

order to decide if this may be the case, the author suggests that if the

"Fresnel" term for scattering from the plasma is large (say, greater

We use this terminology to mean the reflectivity that would result if the plasma gradient wore very short compared with the wavelength of the electromagnetic wave. Section 4 and the Appendix further clarify this term. '

■*—T

-*> -4 than 10 " to 10 ), yet appllcatlor of the gradient degrades this to

-10 less than, say, 10 , then some oxher scattering mechanism Is liable

to be more important than the coherent reflection from the plasma

gradient. An example would be scattering from turbulence or plasma

waves that produce density fluctuations within the plasma gradient.

This scattering is best calculated using Born approximation and a sto-

chastic description of the dielectric fluctuations. Furthermore, under

these same conditions of large Fresnel term, Incoherent backscatter from

free electrons will begin to be as intense as coherent effects due to

the mean plasma structure at power reflectivity values below the order -20

of 10 For this reason, computer plotted data presented here do not -20

extend below 10

L *m ^ J

!

2 BACKGROUND

The complex dielectric COT stant of a plasma is described approxi-

mately by the Appleton-Hartree formula. Under the assumption that

magnetic field effects are small, this dielectric constant is given by

c =1 - rriz (1)

where

X = (f /f)2

P

f = Plasma frequency in Hertz

f = Frequency of the electromagnetic wave in Hertz

Z = v/üü = f /f c

v = Collision frequency in radians/second

uu = 2nf

f = Collision frequency in Hertz.

The plasma frequency is related to electron density through

f = 8.97 • 103/F" Hertz (2) P e

where n = electron density in number per cubic centimeter.

In the earth's atmosphere, Shkarofsky3 has shown that the quantities

Z and X used in £q. (1) are not quite as we have defined them. The

energy dependence of the electron collision cross section with nitrogen

is such that one must use effective values for X and Z that are related

to our definitions in a complex way. Figures 1 and 2 present curves

for relating our definition of X and Z to X ,. and Z ,, to be used in eff eff

calculations. For the data of Sec. 3, we have not made this correction,

References are listed at vne end of this report

■A. . - ^

01 10

■« HIGH ALTITUDE

FIG. 1 CORRECTING Z TO Zeff FOR USE IN APPLETON-HARTREE FORMULA-PROCEDURE AFTER SHKAROFSKY (Ref. 2). Air Slightly Ionized.

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though the effect, in a practical sense, Is very minor. The reader may

do as he wishes in using the data of Sec. 4.

To aid the reader we present in Figs. 3 and 4 our interpretation of

the proper olectron-neutral collision frequency as a function of alti-

tude in the earth's atmosphere. In Fig. 3 the collision frequency is

presented in radian measure. In Fig. 4 the collision frequency is pre-

sented in megahertz in order that the reader may more readily compute Z

for his radar frequency. Equation (2) is plotted in Fig. 5; this curve

gives the plasma frequency in megahertz versus electron density, which

is needed to compute X.

The calculations to be presented assume that the electron collision

frequency is constant throughout the plasma gradient. This situation

pertains to a plasma in which the electron-neutral collision frequency

is larger than other types of collision frequencies. The electron-icn

collision frequency is proportional to electron density. Therefore, as

the electron density increases, electron collisions with ions may be-

come as large as the electron-neutral collision frequency. In the plasma

density gradient, then, electron-ion collisions may begin to dominate the

local propagation parameters. Our analytical results could be in error

under these circumstances.

Figure 6 presents the electron-ion collision frequency versus elec-

tron density for three temperatures of air. Along the right-hand ordinate

wo have indicated altitudes from Fig. 3 at which the electron-neutral

collision frequency attains the same value as the ordinate value. The

reader must use care in employing the results to be presented later,

particularly for high-frequency radars. That is, if the electron density

in his plasma profile—for his air temperature—produces an electron

collision fre^unncy that equals or exceeds at some depth the electron

neutral collision frequency, then he must attempt to determine whether

penetration of his waves to that depth significantly affects the re-

flectivity to be expected. The reflectivity may be increased or de-

creased by this effect, according to details of the situation. We have

not studied reflections from electron-density profiles In which collision

frequency varies with position,

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FIG. 3 ELECTRON-NEUTRAL COLLISIONAL FREQUENCY vs. ALTITUDE

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FIG. 4 ELECTRON-NEUTRAL COLLISION FREQUENCY EXPRESSED IN MEGAHERTZ

10

^ -A.

ELECTRON DENSITY No/cm3

10,000

i 5

Z

g < s

1000

10° 10'

ELECTNON DENSITY No/cm3

FIG. S PLASMA FREQUENCY vs. ELECTRON DENSITY

11

10' 10 ELECTRON DENSITY-

FIG. 6 ELECTRON-ION COLLISION FREQUENCY vs. ELECTRON DENSITY

12

tm fc MZ*.

The data to be presented were calculated for plasma gradients that

are expressed in terms of simple, analytic functions. It is hardly ex-

pected that real plasma gradients will take on the forms that we use.

We suggest therefore that our data be used primarily as a guide as to

whether a coherent plasma interface reflection or some other reflection

or scattering mechanism dominates a given measurement. We also suggest

that if the plasma profile expected is very poorly approximated by our

analytic expressions but the plasma frequency inside considerably ex-

ceeds the radar frequency, then the reader should match our analytic

form to his plasma gradient at a position in the gradient where the

radar wavelength in the plasma is longest compared with the plasma

gradient. We can show that the radar wavelength in a plasma is

2 ' -r—T-H !r 2 21* : (1 - X) + z f 1 + z

1 + z2 J 1 +

- x

z2

where

>. = Radar wavelength in plasma

\ = Radar wavelength in free space.

The wavelength takes on its greatest value at X = 2, if Z is finite.

If Z is zero, then the power reflectivity is 1.0 regardless of plasma

profile; here, profile details would be of concern only if the absolute

phase of the return signal were of interest. The limit of \ with Z = 0

is never mot in practice, however, though some environments do approach

this situation. Therefore, in a situation where the reader encounters

a physical plasma profile very different from our analytical forms, he

might match our analytic gradient at the position where X = 2 (f = /2 f)

On the other hand, our intuition suggests that penetration of the wave

to the X = 2 depth may not be important, so that the gradient matching

should bo done at the X = 1.0 depth, particularly for very small values

ol Z. The choice for matching is left to the reader.

The main effect of electrons in the region where X < 1.0 is probably

wave attenuation. Therefore, should the electron density prolile In this

13

region be very different from our analytic forms, the reader may still

wish to locally fit his physical gradient to our analytic gradient in

the region from X = 1 to X = 2. He would then compute by WKB approxi-

mation the difference in absorption in the region where the profiles are

very different, in order to obtain a more appropriate reflectivity.

Perhaps the most useful application of the data presented here would

be to check results of computer programs that are designed to determine

plasma reflectivities using a layer-by-layer numerical mock-up of the

physical environment. One extreme danger of the mock-up procedure is

that the amount of computer timt> required increases with the number of

layers considered. Since the total number of layers that can be handled

may be limited by computer capacity, the layer thicknesses tend to be

picked too large. As a result, at each interface between layers (or be-

tween shells for a sphericrl geometry), the reflectivity may be larger

than the total reflectivity of the medium. The person performing this

kind of calculation then expects that, since he will be preserving phase,

reflectivities from all the layers will tend to phase-cancel properly

to leave the correct residual reflectivity. The difficulty with this

procedure is that the reflectivity of one interface is likely to domi-

nate, while all other interface reflectivities simply produce a sort of

random walk away from this dominant interface value.

Our analytical results can be used to check such computer programs.

The procedure would be to mock-up our analytic form, layer by layer,

compute reflectivities, and then compare results. Good agreement would

help develop confidence in the layer-by-layer mock-up procedure. Then

the mock-up procedure could be applied to both plasma gradients and

collision frequency variations that cannot be handled analytically.

14

- - *- --*- ■ II ^ !■■

i 3 REFLECTION FROM AN EXPONENTIAL PLASMA GRADIENT

The mathematical derivations of reflectivities used in this report

were taken from Budden .3 Section 17.2 of Ref. 3 derives the reflectivity

to be expected in a plasma gradient in which the electron density varies

exponentially. The derivation assumes uniform collision frequency. We

define

z/z n (z) = n e (4)

e o

whore

n (z) = Electron density at position z

n = Electron density at position z = 0 o

z = Position in gradient (-« < z < +0°)

z = Length over which the electron density changes by a

factor of o. This value would be matched to the

reader's problem.

In terms used in the previous section, Eq . (4) becomes

z/z X * e 0 . (5)

For normal incidence, Hcf. 3 gives, for the power roflection coefficient

[Eq. (17.17) of Ref. 3J,

2 ( 81,Z 1 ) |R| = exp - —^ tan Z . (6)

I ) Figures 7 and 8 are to be used to evaluate Eq . (6). Figure 7 is a

transparency to be used as an overlay on Fig. 8. The ordinant of Fig.

8 gives the power that 10 must be raised (always negative) in order to

obtain the reflectivity.

15

10,000 MHz

Z0 —

7777 6000 MHz

lOOO'K

/JJJJ. i 7TT7

3000 MHz-y l000oK /

FIG. 7 OVERLAY FOR USE WITH FIG. 8 FOR CALCULATION OF REFLECTIVITY FROM EXPONENTIAL PLASMA

16

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10

FIG. 8 SLIDE RULE FOR CALCULATING REFLECTION COEFFICIENT FOR EXPONENTIAL PLASMA GRADIENT

I

1000

100

I IN

UJ

100

105

i

17

An acetate version of Fig. 7 (in pocket under back cover) is over-

laid on Fig. 8 so that the arrow labeled z is adjacent to the gradient o

distance of interest. For an exponential gradient of that value, the

reader obtains the reflectivity exponent for any altitude or frequency

ol concern. As an example, let the exponentiating distance be one meter;

then at an altitude of 10 km, the power-reflection coefficient for a -53

1000-MHz radar wave from a very overdense plasma is, approximately, 10 ,

clearly a reflectivity of little concern. On the other hand, a 100-MHz -5 8

radar wave will experience a reflectivity value of 10 ' , which may be

of considerable concern.

At an altitude on the order of 40 km, at a plasma frequency of

10,000 MHz and a plasma temperature of 1000oK, the electron-neutral

collision frequency and the electron-ion collision frequency are com-

parable. At higher altitudes than this the electron-ion collision

frequency is greater than the electron-leutral collfsior frequency.

Therefore, for 10,000 MHz, a plasma temperature of 1000 K, and altitudes

above 40 km, the reflection coefficient is not given by Eq. (6). We

have shown a shaded region that we believe probably contains the proper

reflectivity value; dt this writing we are not prepared to provide a

precise value.

Similar shaded regions are given for 6000 and 3000 MHz, and 1000 "K

temperatures. For higher temperatures, the ^lectron-ion collision fre-

quency is lower, so that greater reflectivities will be obtained. We

have chosen 1J000K for this discussion because air temperatures would

generally have to be at least this high in order to maintain electron

densities by radiation (or thermally) that are overdense to frequencies

of 3000 MHz and above.

1«

4 REFLECTION FROM A GRADUAL GRADIENT TO FINITE ELECTRON DENSITY

In this section graphs are presented that give the power reflection

coefficient for electromagnetic signals normally incident upon a plasma

cl finite electron density. The electron density in these cases varies

from a value of zero per cubic centimeter very far from the plasma to a

value of n deep inside the plasma. The analytical form for the varia-

tion of electron density is

n (z) = e

1 + e ■z/a

(7)

The parameter a characterizes the steepness of the plasma gradient.

Vacuum exists at z = - as; plasma with electron density n exists at

z = -»• o». For very large negative values of position z,

n (z) *• n e e o

+z/a (8)

This analytical form is like that of Sec. 3. Should n be large enough

so that the wave does not penetrate to a depth where Eq. (8) is a poor

approximation to Eq. (7), then results of this section will reduce to

z of Sec. 3. o

those of Sec. 3 with a

The plasma dielectric constant in the plasma gradient is given by

e(z) 1 - 1 - 1Z

1 + e -z/a (9)

The collision frequency for this form is constant as a function of posi-

tion in the plasma gradient. This condition approximates plasmas In the

lower atmosphere. The power-reflection coefficient for this analytical

gradient form, given in Ref. 3, is

R = i +vq Hi + ikc(/c^ + 1)]

Hi + ika(/^ - 1)] (10)

19

whore i is the dielectric constant from Eq. (9) at z equal to plus

infinity. The parameter k is equal to 2TT/X. The first factor on the

right-hand side of Eq, (10) is the familiar Fresnel equation for power

reflection from an abrupt interface between vacuum and plasma. For this

ruason we refer to this factor as the Fresnel term in this report. The

second factor, involving the gamma function ratio, introduces the effect

of the change in electron density occurring over a finite distance. It

is clear that if a goes to zero, meaning an infinitely sharp transition,

then Eq. (10) is left with only the Fresnel term, as is necessary. The

Appendix indicates in detail how Eq. (10) was evaluated and the accuracy

that we expect in our plots.

The power-reflection coefficient is plotted as a function of Z,

f /f, and a/>- in Figs. 9 and 10. In Fig. 9(a) through 9(m) the logarithm

of the power-reflection coefficient is plotted versus the logarithm of

the ratio j/\. Each graph page is for a fixed value of Z for values of -3 +3

Z from 10 to 10 . The power-reflection coefficient was computed and

plotted versus o/X for a fixed f /f ratio, where f is the plasma fre- P ' P

quoncy and f is the radar frequency. The choices of values for this

ratio that were plotted were made for each graph so as to minimize gaps

between adjacent curves. Thus, different choices for the values of f /f P

will appear on various graphs. The power-reflection coefficient was 2

computed for Z and a fixed value of X, which is (f /f) . The comiuta- P

tion was performed for values of a/X that were stepped in the ratio of

(10) ; in this way, ten points were plotted at even intervals for

each power-of-10 change along the logarithmic abscissa.

The general behavior of all of these curves of power reflectivity

versus }/>., as presented in Fig. 9, is similar. The power reflectivity

is constant at the Fresnel value as a function of cj/\ for small values

of a/X. Then the reflectivity proceeds to decrease extremely rapidly

as the '/>. ratio continues to increase. For fairly low values of Z and

large values of o/X, we note that the power reflectivity changes from

almost no reflectivity for f /f "** 0.9, to a rather substantial reflec- P '

tivitv for f /f •* 1.1. This extreme change in reflectivity with f /f P P

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for large values of oA aids In producing relatively unambiguous lono-

grams obtained with ionospheric sounding equipment. If, for instance,

the ionosphere produced a very sharp Interface (a/\« 1.0), then the

power reflectivity would begin to decrease slowly with an increase in

frequency above the layer critical frequency so that the observed, abrupt

termination of reflection at F f , for example, would not be obtained.

The second group of figures. Fig. 10(a) through 10(m), presents the

power reflectivity plotted versus Z for s variety of values of o/\. The

power reflectivity displayed in this manner demonstrates some interesting

behavior. Most startling is that demonstrated in Fig. 10(g) for the

situation with aA = 1.0. For a value of f /f = 0.9, note that the P

power reflectivity Increases very gradually as Z Increases rather than

decreasing as one would normally expect! The author hopes that this be-

havior results from a physical effect rather than from an as yet un-

realized difficulty with his analytics and approximations. We note that

a similar behavior occurs to a lesser extent in Fig. 10(f) for a oA

ratio of 3 X IO"1.

Assuming that, indeed, the results are correct and the reflectivity

does increase slightly with an increase in Z, we then believe that the

slight increase results from changes in effective radar wavelength.

Thus, at the proper collision frequency, the radar wavelength is "in

resonance" with the plasma gradient. That is, the currents induced by

the incident signal are phased Just right so that their reradiation from

the regions of steepest curvature in dielectric constant reinforce in

the backscatter direction. At higher collision frequencies, the wave-

length in the plasma is too short. At lower collision frequencies, the

wavelength in plasma is too long.

For more shallow gradients (larger aA ratios), f /f ratios would

have to be larger than 0.9 in order for the effective wavelength in the

plasma to be long enough to be "synchronous" with the spacing between

regions of maximum dielectric curvature. In these less steep gradients,

though the signal wavelengths may be nearly In resonance for f /f close

to 1,0, absorption effects could dominate over resonance. For aA ratios

73

less than 1.0, the spacing of regions of maximum curvature say be leaa

than the free-apace radar wavelength, ao that no reaonance can occur

(wavelengths only increase in plasmas) .

Our Intuition tells ua that the observed increaoe in reflectivity

will be mathematically observable only for plasmas with o/X value« near

1.0 and f /f ratioa nearly equal to, but leaa than, 1.0. We auapect

that the reflectivity valuea in Fig. 10(g), where thia "reaonance"

appeara, are ao very low that the coherent reflection from the plaama

Interface phenomenon that we are studying in thia report Is unimportant.

Therefore the reaonance phenomenon ia of no practical importance.

The second interesting behavior exhibited in Figs. 10(e) and 10(f)

Is the "plateau" in reflectivity at large valuea of Z and at valuea of

If that are very large compared with 1.0. We explain thia phenomenon

as follows. For fixed u/\ value, very amall Z, and very large f /f

(overdense, collisionless plaama), the reflectivity takes on the Freanel

value—which is near 1.0. The radar-wave reflection occura primarily

In the region where t /t m 1,0, As the collision frequency is increased,

the reflectivity value begins to decrease as the collision-produced real

part of the propagation parameter allovs energy to propagate into the

overdense plasma region. For very large f /f valuea, the reflectivity

value will decreaae as Z (the electron cdliaion frequency) increases

with values that are the same as those computed in Sec. 3.

As Z further increases, the plaama continues to appear as an expo-

nential plasma as discussed in Sec. 3 with the arctangent Z term of Eq.

(6) saturated near TT/2; further increases in Z cauae no observable change

in arctangent Z. Thus, the reflectivity value levels off as a function

of increasing Z. In the plasma, as Z is increased, the effective region

where reflections occur penetrates deeper and deeper Into the medium

even though the power reflectivity value remaina constant. Eventually,

for a largo enough value of Z, the wave penetrates deeply enough into

the plasma so that the Eq. (8) approximation to Eq. (7) la not valid

where reflection effectively takes place. That la, for the exponential

plasma of Sec. J the region of effective reflection occurs at a depth

74

-i£fe.

J» ' ■■■■'■ ■ ^

I

where the electron density exceeds lhat which our particular finite-

density plasma achieves. Therefore, our finite-density plasma becomes

ineffective as a reflector, so that further Increases In Z are accom-

panied by a decrease In power-reflection coefficient. This Intuitive

explanation for the observed plateau values for reflectivity Is supported

by the fact that reflectivity values for the exponential plasma of Sec. '

3 agrees with the plateau values. Figure 11 presents plots of reflec-

tivity for the profile forms of Eqs. (7) and (8),

75

A --^

10 100 COLLISION FREQUENCY TO RF RATIO, Z,

1000

FIG. 11 COMPARISON OF POWER REFLECTIVITY vs. Z FOR A GRADUAL GRADIENT AND AN EXPONENTIAL PROFILE

76

5 CONCLUSIONS

In this report we have presented data to allow the reader to esti-

mate the power reflectivity of an electromagnetic signal normally inci-

dent upon a plasma gradient. The electron collision frequency is assumed

constant throughout the plasma. We have provided graphs that allow for

simple determination of plasma characteristics in the atmosphere. We

have presented in Sec. 3 a "slide rule" to allow easy estimation of the

power reflectivity from an exponential plasma (electron density goes to

infinity) in the earth's atmosphere. In Sec. 4 we have presented data

for a specific analytical form for the plasma gradient to a finite,

uniform electron density.

For reflection from atmospheric plasma gradients, we suggest the

followinft procedure. Determine whether or not the atmospheric plasma

is overdense to the electromagnetic signal of concern. If it is, then

use the slide rule of Figs. 7 and 8 of Sec. 3 to determine the power

reflectivity—provided the form of the gradient is well matched by the

exponential form. If the reflectivity is large enough to be of concern,

then the reader would be well advised to pursue the problem In some

additional depth by means of the data presented in Sec. 4. The power

reflectivity as given in Sec. 3 will always be larger. If -he reflec-

tivity is very small [less than 10 say], then, as indicated earlier

in this report, some other reflection mechanism may be more important

than the coherent effect of the plasma gradient. The reader would then

be well advised to study other possibilities.

If the atmospheric plasma is not overdense to the signal of con-

cern, then the data of Sec. 4 must be used. Again, the reader should

bo alert to the physics of his environment inasmuch as mechanisms other

than smooth plasma gradients can reflect electromagnetic signals.

In either case, overdense or underdense plasmas, the reader can

probably match well enough the plasma gradient in the reflecting region.

Uilforences in plasma profiles in the exterior can be handled by comparing

77

Signal absorption by WKB approximation for the twc external regions

Appropriate corrpensation can then be made.

78

L mm fc. A

Appendix

REDUCTION OF ANALYTIC FORM FOR REFLECTION

FROM A GRADIENT IN PLASMA DENSITY

79

Appendix

REDUCTION OF ANALYTIC FORM FOR REFLECTION

FROM A GRADIENT IN PLASMA DENSITY

Reference 3 (Eq. 17.112) gives for the voltage reflectivity between

a vacuum and a plasma the following form:

(- 2ikoC) C - Qc

v " C + q- (2ikoC)

): r[ika{q2 + C)}:"!

T " [likaCqg " C)j!j (A-l)

where

C = cosine of incidence angle (C = 1 for normal incidence)

2 q = e2 complex dielectric constant deep inside plasma

e2 =1 - r^iz with X and Z defined in Sec. 2, and

k = 2TT/X

X = Radar wavelength

0 = Distance characteristic of change in plasma density [see

Eq. (A-2)].

The functions indicated are factorial functions that are also gamma

functions. They have complex arguments.

The expression (A-l) gives the voltage reflection coefficient (in-

cluding phase) from a plasma gradient in which the dielectric constant

varies in the following way:

e(z) =1 - 1 - iZ ,. -z/o, (1 + e )

(A-2)

This form means that the collision frequency Z is not a function

of position, but X is. In fact

n (z) = n e o 1

(1 + e-*")

81

(A-3)

For large negative values of z we find that the electron density

is nearly a simple exponential as was used in Sec. 3. For very large

negative values of z, the dielectric constant approaches the constant

value 1. At very large positive values of z the electron density

approaches a constant value of n and the dielectric constant similarly

approaches a constant value.

We shall be substituting a value 1.0 for C (wo are only interested

in normal incidence). The first factor on the right-hand side of Eq.

(A-l) we call the Frosnel term. This factor gives the plasma reflection

coefficient should the plasma gradient be exceedingly abrupt compared

with the wavelength.

The second factor on the right-hand side of Eq. (A-l) has modulus

1.0 and only affects the phase of the returning signal. We shall be

concerned only with power reflectivity, so that this factor is of no

concern. However; as an exercise, for real o (the only kind wo consider)

2 (- 2iki) : (+ 2ik-i) :

TU - 2ikj) Hi + 2ika)

1(7) (A-l)

with z here meaning the complex number (1 + 2iko) . If we apply the

ident ity

ru) - rm (A-5)

we f ind

rm r(z)

r(F)r(7) r(z)r(z)

r(z)r(z) _ , .. rmrrr) = lü (A-6)

An evaluation of the third factor on the riRht-nand side is not

straightforward. We find

[[lko(q2 ♦ n]:"|2 [TU * tko(q2 * Dl"!2

fik(q:, - in:J |r[l ♦ lko(q2 - ml (A-7)

H2

.A. m.*.

We let

Qo = M - iX (A-8)

and find

[Hi -t- kox + ika(n + 1)]V

with

[Til + kox + iko(M - DlV

x = 1 + kax

[fCx + iy^]'

[r(x + iy2)]' (A-9)

(A-10)

y. = ko(M + 1) (A-ll)

y2 = ko(n - 1) (A-12)

Reference 4 [Eq. (6.1.25), p. 256] allows us the following trans-

formations :

or

FCx + iy^

rex) r(x + iy2)

r(x)

/1 n=0 1 t

2 yi

(x + n) CO

IT 1

/I 2 y2

(x + n)'

2 \d

[ko(M - DJ /

n_0 -JD

n=0 ( In ♦ 1 ♦ k-xl2 )

(A-KO

(A-14)

Wc have found that the evaluation of the continued product of Eq

(A-ll) may require tuns of thousands of factors before the continued

product convurges to within a low percent of the proper answer. We

83

J

later develop a technique that combines continued product evaluation and

analytical approximation to obtain an accurate representation of Eq .

(A-14) using a very reasonable amount of computer time. We return to

this shortly.

2 The determination of M and x from q is fairly straightforward. We

find that by defining

C = 1 r

1 + Z (A-15)

and

and

XZ

M = +

X = +

1" I 1Z2

>

V^r 2

f «i + er 2

Vv 2 ' ei " er

(A-16)

(A-17)

(A-18)

In terms of u and x, the first factor on the right-hand side of

Kq. (A-l), when absolute squared, becomes the very familiar

2 2 _ (i - y * x F "' .2 2

(A-19)

(1 + M) + X

which is the power-reflect ion coefficient for an abrupt interface be-

Iwuon the plasma and a vacuum. This is known as the Fresnel equation

and wc call it, hero, the Fresnel term.

Wo sue from Kq. (A-l) and (A-14) that should the parameter j

approach zero (an abrupt interface), then, since Lq. (A-14) takes the

value 1.U, Kq . (A-l) does reduce to the expected form.

Hi

Because of the slow convergence of Eq. (A-I4) we have developed an

analytic approximation to evaluate the continued product. In our compu-

tations we have usually used a hybrid form, as will be described. We

define

a = ka(M - 1)

b = ko(M + 1)

(A-20)

(A-21)

and

y - m + kov m * (A-22)

Then Eq. (A-14) becomes

/ 2 2 v ym ^ a | n

m=l yJm 2 K2 ( y_ + b )

(A-23)

or

or

P - exp (log TT [ ]2 = exp h V^ log [ ]

m=l I I m=l

P = exp {2 Y2 l0ge (y» + a2) 'X] l0ge (y« + b2) m-1 m=l

Translorming the summations to integrals we find

_a>

P = exp '2 / log I y_ + a |dm - / K

2 ^ni=l/2

(A-24)

(A-25)

!B CD

/log ym + a dm - / -og y^ * b dm

=1 2 Jm*l/2

(A-26)

85

The integrals may be evaluated to give

P = oxp/2 TT(a - b) - y log ID O

2 2 y + a jn 2 K2

■

21 w & _—1 I" "l b tan — - a tan «

(A-27)

We have freedom in the way we may use this analytical expression

in that y , the lower limit, can be chosen at will. Thus, if we wish m '

to use Eq. (A-27) as a complete substitute for the continued product,

then

5" + kox (A-28)

Wo use the 1/2 in Eq. (A-28) since the value for m = 1 of the con-

tinued product is approximated by the integral from m = 1/2 to m = 3/2.

Test cases run on computers have shown that Eq. (A-27) may be in error

as a complete representation of the continued product of Eq. (A-14) by

as much as 10 percent, but usually, particularly for total reflectivities -20

larger than 10 , has an accuracy exceeding one percent. Greater

accuracy can be obtained by evaluating the first m factors of the con-

tinued product, and then multiplying these by Eq. (A-27) with

ym - m + 2 + kaX (A-?9)

Table A-l presents a matrix of values for power reflectivity in

which the hybrid form of m factors of continued product times the

analytic form from m * I/'2 to « was evaluated. From the table we see

that the analytic approximation always gives one percent accuracy for

reflectivities larger than 10

Evaluation of the continued product by Itself more often than not

required thousands of factors to converge to within 10 percent of the

proper limit. Table A-l also presents some results on this aspect. For

all plots presented In Sec. 4 of this report, a hybrid calculation com-

bining the first 10 factors times the analytic expression from (10 ♦ 1 2)

86

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< 2 H u

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t ss u 1—1

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1 |

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Ä 0 a O CO s rH 4-> 3 B 0 n "K 0 «-> B II CD

II 0) 3 B ■H be FH

h '1 0) «-> X in s FH ,<: K C 1 -F4 •H • s U < 0 in «J t e V a & 0 • • 3 in

CM I 1 01 FH

•»-> •o in ]

a m B a •<-> B a V II 3

C •» w n Ü CO K s

r, • i •^ >. IN X TJ •>. h (A J« B ♦J Ü 0 hi C • 1 CD

0 B O t. O » •v •F< 0 ä ■ a 4-> in

(A ♦* b I • Ü B B B ■ tl >. | CO 0 0 6 <M Ü 0 «-> •s ■M

■-< 0 a .J ■F4 3 >. ■ *J

w 0 B B O £ « a IM «) ♦* O •F» •F4 CO 4-> 5 FH 5 «- B rt <M 4-> k. 3 0 Ü ^5 B B 3 •. X o w 0 « ■rt 0 o IA £ FH

tf <J u o 0 u h

a n I F4 0 CD 0 a o •»< lM «-• •M ♦^ a

1 B 0 «■> u C« 0) <M o V B B IS « F-« 0 FH -C 0 £ 0) S-P U 0 F-t » .H K -«4

r 0 u >. 1« 0) rt ■•-• «-' <M 1 FH 0 0

B 0)

♦J

& g >. FH

h 0 § > u ♦4 0 •s CO 0 J= -t h X 8 o o B u. <-> B «-• V 0 FH -* CO

87

*

L ■A ■ ,^

\

to infinity was used. Wo assume that all points are then accurate to ]

within one percent.

The form of the dielectric constant used in Sec. 3 is a special

form of that used in Sec. 4. The analytic form of Sec. 3 is obtained

if in Eq. (A-2) we set o = z and let X go to infinity. As a check on

the validity of our procedures, we used our analytic approximation to

the continued product, allowed X to approach infinity and, indeed, found

convergence to the expression given in Eq. (6) of Sec. 3. The procedure

is tedious and will not be reproduced here.

88

^ --*- ■ ■■ ^ .«■ i

^ p

(

REFERENCES

.), A. Hatcliflc, The Magncto-Ionlc Theory and Its Application to Ihu lonospheru (CambridKe University Press, 1959).

I. P. Shkarofsky. "ücnuralized Appleton-Hartrec Equation for Any Degree of lonlzation and Application to the Ionosphere," Proc. IRE, Vol. 11. pp. 1857-1871 (December 1961).

K. (i. Budden, Radio Waves in the Ionosphere (Cambridge University Press, 1961) . " " " ~ "

ilandljook of MiithL-mat ical Functions, Edited by Milton Abramowitz and Iren«: A. Stogun, Applied Mathematics Series 55 (National Bureau of Standards, November 1964).

89

UNCLASSIFIED 8>cuiUy CU*eUic«Ctow

DOCUMENT CONÔL DlTA R&D (tttutlly claiilllemttnn »/ iltl; body et ■ bund mnd Indttlnt annotmtlon muH bt »nltnd whmn lh» orrrtll npmrl I» tlm»»Hlm4)

I OMieiNATiNO kctiv\ry (I -itpotmi* •mhot)

Stanford Research Institute 333 Ravtnswood Avenue Menlo Park. California 94Q25

M. nlÔNT tCCUMITV t 4ttlFlc«TIOH

UNCLASSIFIED afe. owoup

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RADAR REFLECTION COEFFICIENTS FROM A PLASMA GRADIENT WITH COLLISIONS

4 OCtCnOTIVC NO T ( t rTV»* of r«*or( «nd <nclu««v> <*•!••)

Special Report 10 • «u THOKitl (Fltii namm, mtitdl» Inlllml, !••( nam«;

Walter G. Chesnut

• nKPONT DATC

October 1968 7a. TOTAL NO. OF PASCi

108 7». NO o' ntwt

•a CONTHACT on anaNT NO

DA-49-146-XZ-184 b. nnejccT NO

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Special Report 10 SRI Project 4417

9b. OTHtn KKPOHT NOItl (Any olhtt thlm nport)

numfeara Mat mmy b» fl0)»d

10 OltTKiauTION tTATCMCNT

This document has been approved lor public release and sale; its distribution Is unlimited.

II tUPPLCMCNTaRY NOTES 12. «PON»O"IN0 MILITARY «CTIVITV

Headquarters Defense Atomic Support Agency Washington, D.C.

i* «•«rnacT

This report presents numerical results in the form of graphs of the power reflection coefficient for electromagnetic signals normally incident upon a plasma gradient. An electron collision frequency that is Independent of position in the plasma interface is assumed. A "slide rule" l'j provided to enable quick determination of the magnitude of the reflection coefficient for "very overdense, exponential plasmas" in the lower atmosphere (to 70 km) for frequencies from 30 MHz to 10,000 MHz. For plasmas that only reach a finite density, a complete set of graphs is given presenting power-reflection coefficients for normalized plasma parameters. Reflec- tivities are plotted if greater than 10"2 . Normalized collision frequencies (Z) from 10" to 10+ are covered. Normalized plasma to RF frequency ratios inside the plasma cover the range from 0.01 to 100 (10 '• X » 10+ ). Characteristic distances over which the plasma density exponentiates in the gradient region vary from 10 w&velongths to 10+J wavelengths. Simple aids to the reader are provided so that he may easily determine his parameters for use with the reflectivity curves, should his plasma be of atmospheric constituents.

DD 'r.1473 mffiuaamjEB Secunlv CU^silu alion

UNCLASSIFIED S>cufity CUi>»ic»ttöir

Kir wo Mo* LINK A LINK • LINK C

ROLB NT

Radar Reflection

Plasma

Low Altitude

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RADAR REFLECTION COEFFICIENTS FROM A PLASMA GRADIENT … · Normalized plasma to RF frequency...

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