National Bureau of Standardf

Library, N.W. Bldg

MAY 2 2 1964

^ecknlcai viote 92c. 2//

Vohune 5

CONFERENCE ON NON-LINEAR

PROCESSES IN THE IONOSPHERE

DECEMBER 16-17, 1963

EDITORS

DONALD H. MENZEL AND ERNEST K. SMITH, IR.

Sponsored By

Voice of America

and

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NATIONAL BUREAU OF STANDARDS

technical *9tote 2//, Volume. 5

Issued April 24, 1964

CONFERENCE ON NON-LINEAR PROCESSES

IN THE IONOSPHERE

DECEMBER 16-17, 1963

Sponsored by

Voice of America

end

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D.C. 20402

Price 60*

Volume V

Table of Contents

Session 3. Collisional Eadlo-Wave Interactions. I^rt II

3.7 Electromagnetic vave reflection from an oscillating

collision-free magneto-ionic medium, O.E.H. Eydbeck,

Research Laboratory of Electronics, University of

Technology, Gothenburg, Sweden

Session h. Excitation Processes.-

Chairman: V. A. Bailey

4.1 Non-linear interaction coefficients for electrons in

nitrogen and air, A. V. Hielps, Westinghouse Res. Labs, . ^9

4.3 The excitation of optical radiation by high power density

radio beams, Lawrence R. Megill, CRPL, EBS 69

h.k Alteration of the electron density of the lower ionosphere with ground based transmitters, Pietro P. Lombardini, University of Pennsylvania 87

4.7 Nonlinear effects in radiation generation through the

coupling of electron beams with diffraction gratings,

Winfield W. Salisbury, Varo, Inc., Garland, Texas. . . Ill

X-Session 3 paper continued from Volume IV, Technical Note

No. 211,

X"X' . Session k papers continued in Volume VI, Technical Note

No. 211.

Xll

ELECTROMAGNETIC WAVE REFLECTION FROM AN OSCILLATING,

COLLISION- FREE MAGNETO-IONIC MEDIUM

O, E. H. Rydbeck

Research Laboratory of Electronics

Chalmers Institute of Technology

Gothenburg, Sweden

1. Introduction and Summary

A collision-free or low collision magneto-ionic medium is

easily perturbed by a driving or pumping wave, even of moderate power.

Unless the pump wave has longitudinal propagation, it will generate

longitudinal electron velocities, as well as differential space charge

densities. These quantities, as well as the transverse, pump field

electron velocities, will affect the propagation of any (low power) signal

or probing wave propagating through the medium, which now has

oscillating and travelling electron velocity and electron density ripples.

If the pump wave has an angular frequency co , and that of the

signal wave is co , two first order sum and difference frequency

(to =00 ± CO ), waves are generated in the medium. It is shown that ±1 p s the generation of these waves, for which specific refraction laws hold,

is greatly enhanced if a parametric travelling w^ave resonance develops

in the system or if the non-linear driving force experiences a local

resonance. Under certain conditions, sum and/or difference frequency
waves, usually with tw^o kinds of polarization, will radiate away from

the resonance interaction region. This generation is dealt with in detail

for homogeneous media, with a discontinuous boundary, and for slowly

inhomogeneous ones.

Waves of this kind should be generated easily by a topside

sounder of moderate power. It should also be possible, with present

day techniques, to record sum and difference frequency "echoes" at the

ground, if a very powerful pump wave transmitter were used. It is

interesting to note that not only the true height of normal reflection

2 2

(N_e /m € ^ = CO ) but also the electron density gradient, at the same

s

level, should be obtainable by such measurements, provided

CO :^ CO , Since co could be left unchanged, while co is swept

p s p s max

through the sounding range of interest, the technical arrangements for

such experimients, which might be very rewarding, should not be overly

complicated.

2. Electromagnetic Wave Reflection from the Oscillating,

Isotropic, Ionized Medium

a. Only electron density oscillations considered. Let us

assume that two plane electromagnetic waves, one signal wave, of

angxilar frequency co and one high power (pump) wave, of angular

frequency co impinge upon a plane, homogeneous and isotropic as
sho'wn in figure 1. We assume that the pumpwave is powerful enough

to produce a differential charge density, AN, sufficiently large to

affect the propagation of the much weaker signal wave. The influence

of the electron velocities generated by the piimp wave, which affects

the propagation of the signal wave in a similar manner, will be considered

in the following section b.

The differential electron density can be written

AN = T] N cos (co t - k 7) , (2. 1)

p p

where

k =^ = -^n (2.1a) ^

p ^

and X. is the wavelength of the pump wave in medium 11.

The wave equation for the signal field becomes

a

2 +

a

2

1 a 2^

"0 . — 2 " ~1 2 2 (1 + ^cos(co t - k r))^|j =0, (2. 2) az ax c at c ^ ^

where \1> is related to E by the relation

y

E =^ 1^. (2.2a)

y

jo)^ at

If we neglect the field instabilities generated at or very near the

parametric system resonances (for details see Rydbeck [1963]), E

can be written
E = E

y y

X ) z z ^z

e < e + a e +

-1

-j(k - +k )z ] +m ,t-k , x) ''^ -1 p

* -"^ +1 +1 'Ui , + a , e +•••?+€ ^a, , e -'+1

1 '"s 1 *^

- J k , z

-^ +1

1

1

-j(k^ +k )z

^ +j{^_^t-k_^x)

^ -jk_j z

, zzl x-1 Z + a e +---We — a_^e +

+1

J s

-j(k -k )z

s p

z z + higher order frequency terms,

(z^O). f2. 3)

where kSo, k are the x and z components of k , and

S X z

UU / OJ uu k = — jl - -^ = — n(a)) (2.4)

(c = the electromagnetic velocity in vacuum) .

Furthermore, uu and uu denote the sum and difference frequencies; i. e. ,

and

UU , ^^^ = UJ ±uu , uu_ ^

/ s p

^1 ^-1

^l="^"^l^' ^1 =^-tu).j)

(2.5)

(2.6)

One finds that [Rydbeck, 1963] that a and a can be written

» +1 -

a / 2 2 2 ,_ _ ,2 • (2.7)
These amplitude coefficients exhibit parametric travelling

wave resonances and approach ± » (losses neglected, also possible

instabilities at or near these resonances [Rydbeck, 1963]), when

^l^^'^+^|

(2. 8)

k , ^ ± k -k

-1 ' s p

Whether these resonances develop or not depends upon the boundary

conditions which determine a . a . the amplitude coefficients of the

natural, or unperturbed, sum and difference frequency waves of the

medium. Since a . , a are proportional to these coefficients

-1 +1

times n, they will be neglected in the following analysis (a and a

+ 1 °-l

will be proportional to r| ),

The boundary conditions at z = require that

k + k = k , and k - k = k , , (2. 9) s p +1 s p -1 ' ^ XX ' X XX X

which yield the following reflection laws (in medium I) for the "reflected"

s\im and difference frequency waves, as shown in figure 2,

viz. , sin cpj = ^ ^ ^

{iu^ sm Cp^ ± uu sm cp^ ) . (2. 10) + 1 s p s p

-1

In the case of degenerate parametric pumping, i. e. , uu = 2uu , this

P s relation yields the peculiar result

sin cpj = -,sin cpj (u) = 2()u^) . (2. 11)

-1 °
If higher order sum and difference frequency terms were

considered, one would obtain the more general relation

1 sin Cp (UU sin Cp_ ±miu sin Cp )

,

(2.10a)

I uu±mtjjss I„i p 1 +n s p p

-n where m = 1, 2, . . . .

If we next introduce the boundary reflection and transmission

coefficients of the unperturbed signal wave, fiz. ,

R

S S z z ='°%° +k s s

2 k

; T

''' k^H-k

s s z z

(2. 12)

where k = uu /c , the boundary conditions finally yield the following

z

field relations in media I and II (E = amplitude of incident signal wave):

Medium I (z ^ 0)

E = E

y y

r+j(uut-k x-k z)

s s s X z „ e + R ^ e

s,

+ i(UUt-k x + k z)

s s s X z

^^^ +J(Ui^,t-k°j^x+k°^^z)

+ 1 s

-1 s

(2. 13)

where k = (JU /c , and k = ud ,/c . The sum and difference

"reflection" coefficients, R and R , become
R

k , - k - k

+ 1 s p z z z °'+^ k° + k

+1 + +1

z z

(2. 14)

R

k , - k + k

-1 s p

z z z

0. -1 ,0

z z

(2. 15)

Medium II (z ^ 0)

E = E° T ^ y y s,

+ j(ujt-k x-k z)

s s s X z

+ 1 X J z z z + OL^ e <e -Te f + \l% 1 O.+l

^

+j((^.lt-k_j X) j^ -j{k^ -k )z -jk_^ z^-1

-1 xj z^z^ z

w^here

k

.° + k + k

+ 1 s p

_ z z -^z

+ 1 ^ +1

z z k ° + k - k

-1 s p

z z ^z

z z

(2. 16)

(2. 17)

(2. 18)

It is interesting to note that, for k =0 (non-travelling pumpwave),

these transmission and reflection coefficients (T ; T ;

R_ ^; R ) are the regular ones for a frequency shifting network.
By (2. 9), relation (2. 7) now can be written

z z z which means that

2

r| UL) 1 1 ^0 +1 % ^ 7 "2 "1 ' <^- 20)

^'"^^ ^1 2 c^ k°. +k . k +k +k 0+1 +1 -Us p z z z ^z

2

-J.) 0) 1 1 %. -1 % = - ^ -5 • <2- 21)

-1 2 c^0-1k , + k , k , + k - k

-1 -1 s p z z z z

Thus, to first order, only two parametric travelling wave resonances

are left, viz. ,

and z z ^ ^2. 22)

k = - (k - k ) . -1 s p

z z

The other two resonances cancel out due to the boundary reflection

effects. Since the energy balance at the boundary requires that the

sign of k be equal to the sign of - (k ± k ) , this resonance is z z possible physically, if only the condition |k | = |k ± k ] can be met

z z

(a condition that is more easily attainable in the magneto-ionic medium).

The natural sum frequency wave in medium II always travels

in negative z -direction, i. e. , towards the boundary.

8
As far as the difference frequency wave is concerned, we

have to distinguish between two cases, uu < uu . If (JU > uu , the ° p s p s natural difference frequency wave travels in positive z direction if

k < k , and in negative direction if k > k When uo < (U p s p s p s ^z z z z

this situation is reversed.

A degenerate parametric resonance, |k | = jk |. This

~ X s resonance therefore is possible only if 2 | k | > k , i. e. , if

n (uu) > n. . If k is determined by the properties of the isotropic

ir ir ionized layer only (for an unguided electromagnetic p\imp wave), this

resonance is not possible for travelling waves.

2 2

If we assxime that k =0 (for example if (U = tu^), the

P

^ p

degenerate parametric resonance requires

^s = -J74 + sin%j = n_^ , z z which yields one exponentially decreasing wave (with z) in medium 11

and one exponentially increasing; together they constitute a negative

energy flow (at the frequency UU = uj /2), which generates the

s p

"amplified" signal returning at the peculiar angle - cp from the

boundary (see (2. 10)). The pumped ionized medium now acts as an

amplifying wall or resonant reflection amplifier, the necessary power

being supplied by the pump wave. Obviously one could consider parametric plasma amplifiers operating on this principle.
In an anisotropic medium, for example, the magneto-ionic

one would have many more degrees of freedom to obtain parametric

travelling wave resonances. Actually, resonance coefficients like

a , (2. 7), are quite general in nature, and similar expressions are

°±1

obtained for the magneto-ionic medi\im, as will be shown in paragraph 4.

b. Electron velocity oscillations also considered. If we, for

the sake of simplicity, assume that the electron velocity wave and the

differential space charge density wave propagate with the same

k -value (this is not always the case), it can be shown [Rydbeck, 1963]

that nin (2. 7) has to be replaced by t] for a. , and ri for

a- , where

uu uu V n . . Tl^l = ri ±^ -^-^ cos {^^^ - q)^ ). (2. 23)

u,^ Cq p

In this expression, v denotes the amplitude of the electron velocity

P

(v) caused by the piimp wave, and '^Ij the angle that this velocity makes

P

with the z-axis.

It is important that (1) the inclusion of the electron velocity

does not change the parametric travelling wave resonances, and

(2) that electrons oscillating transversely to the signal wave normal (in

(v) which case cos (<p|j - cp,, ) = 0) do not affect the propagation of the

p

signal wave, as one might expect for the homogeneous medium. Since

(v) V is proportional to the pump wave amplitude (except when cpj = Cp ) p

10
and T\ is proportional to its square (for an electromagnetic pumpwaves

in an isotropic medium), the electron velocity component normally

would be the most active one in generating the sum- and difference

frequency waves in an isotropic mediiim (see also paragraph 4).

3. Generation of Natural Sum and Difference Frequency Waves

in a Slowly Inhomogeneous Medium

We have seen in the preceding paragraph how the boundary

conditions at z = z determine the possible parametric travelling

wave resonances and the amplitudes of the sum and difference frequency

waves. Naturally, one would like to know what happens in an

inhomogeneous medium, if resonances occur at some specified level,

Z = Zj,.

Let us therefore consider an inhomogeneous medium, but a

slowly varying one, so that partial reflections of the signal and pump

waves can be neglected. Furthermore, we limit ourselves, for the

sake of simplicity, to an analysis of the difference frequency wave.

Since the medium is slowly varying (in the z -direction) the signal wave

z assumes the form exp {j(ujt-k So x-/k dz)}, and similarly, the

/ S X "^ zz pumpwave can be written oos(ajt-k x- k dz) . The natural

P Px J Pz

difference frequency waves are written

11
(1) j((JU jt-k ^ X T f k ^ dz) . n^^> = e

" "x J "z ^3^^

By (2. 2) we now obtain for the difference frequency wave

f =a{n'^'J n'i'5dz-n<')/ n<^'?dz3, (3.2)

z z

1 3

since

k , = k - k , -1 XXXs p

2 j(ou_^t-/k^ <i^+/^p ^^> '

2 c?

and

(2. 2a)

For a slowly varying medixim, a = rrr , and therefore by

•^ _1 z j{uut-k,(x-x)} 2

E ^ E e — :r- - ^- • y. yo % 2J^.1 2 2 z

z + j fk^dz z-j/(k -k +k)dz

^. -1_

p

z s„ p„ -1^ z^ z r ^a 2

^ ^

e

z z -j/ k dz z -j / (k -k -k )dzi

.e

'^ ^ f, \ ^ Pz -^z

^

(3.5)

12
One notices that the two coupling integrals may "generate"

natural difference frequency waves at or near their respective travelling

wave resonance levels, compare (2. 7),

and

k . = - (k - k ) (3. 6) -1 s p

z z z

k , = k - k . (3.7) -1 s p X XX

If, for example, k < k and uu > uu , these resonances actually

p s p s z z yield forward, natural difference frequency w^aves, and backward

(i. e. , "reflected" from the resonance region) waves if k > k . ,p s ^z z

If we assume that (3. 6) holds at some specified level

z = z , we can write as follows for the slowly varying medium:

j(k^ -k +k_^)dz =U{z-z^)^/2 = W^, (3.8) z z ^z z

where Ji = [ ^

(k - k + k , )} . (3.9) dz's p -lz = z

z ^z z

Since we only need to know the electric field of the emerging

natural, difference frequency wave (i. e. , radiated from the interaction

region), we can limit ourselves to such z-regions, or |z-z [ distances,

2

for which W ^=^ 1. If we make use of the asymptotic form of the Fresnel

integral and assume that the condition for forward radiation is

satisfied, relation (3. 5) yields the following difference fields, viz. ,

13
z < z , W » 1

j{a)t - k (x-x ) -/ (k - k )dz}

- -1 az_s p

E = E e X z Pz '"-1

s -1

(3. 10)

z < z , W » 1

E = E e y yn

j { uj t - k , (x - X ) - / (k - k ) dz }

X a z z -1 ,

(JU <a.

1 T] OU /n

+ j( / k dz + n/4)

7. ~ 1

-1

z

(3.11)

Relation (3. 10) is as the difference frequency part of (2. 3), This wave,

which alone does not satisfy the wave equation, is associated with the

signal wave and can be considered as a "modulation" or side -band of

the same. Its resonance is independent of the sign of k - k , as it z z shall be.

/Ztt

It appears from (3. 10) that /-—- is a mieasure of the

difference frequency radiation efficiency of the interaction region. If

H is small, i. e. , if the medium parameters vary very slowly, the

2 radiation is large, provided the region is so extended that W is

large at its outer edges.

If we next consider the second resonance (3. 7), which

takes place at the same frequencies as the former one, (3. 6), one finds

that (3. 5) yields the same result as (3. 9) and (3. 10). The difference

14
between the two resonances thus is purely formal. This agrees with

the results found in paragraph 2 for the discontinuous boundary case.

If k > k , and u) < UU , we obtain a "reflected"

p s p s z z natural difference frequency wave, i. e. , when the associated difference

frequency wave (non-linear driving force) travels in a negative

z-direction. When the opposite conditions hold, we get a forward

difference frequency w^ave. Since the natural sum frequency waves

radiate in the running direction of the associated sum frequency wave,

a "reflected" sum frequency wave can be obtained only if the pixmpwave

runs in a —negative direction with - k > k . This is in contrast to the = p s z z case of the discontinuous boundary (paragraph 2) which always yields

"reflected" sum and difference frequency waves.

The amplitude of the difference frequency reflection

coefficient now can be written

uu , ri uu- 1 / 2iT

Z

which corresponds to | T R | of (2. 13).

The frequency dependent phase of the frequency shifted

"echo" is

"o

^0. -1 J

(k -k -k , ) dz (m >uu ) . (3. 13)

s p -1 p s z z z \ '^—-—

«2(k -k ) "z Pz

15
Since z depends upon uu (as well as uu ), the echo delay, T,

vj s p

counted at the level z for a pulsed signal wave (cp^ and cp^ a

®

I I s p

assumed to be zero) becomes

3 f I^Ki 1 1 V

1 V

g

r k— + T— h (3. 14)

where v and v are the group velocities of the signal wave and

^s g.

the natural difference frequency wave. For a forward wave, emerging

/I 1 ,'. (3. 14a)

V V

at z = z (>z ), one similarly obtains

1 1 I'^^-i

8s 8. \ /^o \ 8s 8.

We thus get a change in effective reflection height of

(k / Jt) which may be a large quantity. It is a measure of the

^z ^0

"frequency shift reflection thickness" of the interaction region. As the

dispersion is very large, it is not sufficient to make use of the first

phase -frequency derivative only to determine the time of arrival of the

frequency shifted wave. Such a discussion lies outside the scope of the

present communication.

and ^ ' ^ '^0

k , = k = k , -1 s p X XX

one can sketch the frequency shifted wave normal paths about as shown

in figure 3.

16
4. Electromagnetic Wave Reflection from the

Oscillating Magneto -ionic Medium

Let us assurae that the static magnetic field lies in the

x-z plane, that its electronic angular gyrofrequency is m with

ri

components uu = uu cos 0, (A) = uu sin 6, along the z and x axes,

1j rl T H

respectively.

If we introduce the following operators, viz.

^ = ^ + t; ^ = ^ + Ut)^ e,,^^2 ; and P = -^ + UU , (4. 1) 2 em e 0' p^20' ' MqSq Bz at ^ Bt

and limit ourselves to the case cp^ = = cp (i. e. , vertical incidence)

""s p

we obtain the following non-linear wave equations for the (second

order) electron velocities, v , v , v , [Rydbeck, 1962 1,

y x z

Dv^^^ = P^ (uu P^ t +P^ -^) - 0)^ P^ P^ i|; = n , (4.2)

y p L e X em T em e z y

o t

B t

Dv^^^ = P^ (UD, P^ S^^^ i|/ + P^ sl^^ _Z ). uu P^ P^ sl^^ t = n , (4. 3) X p

* L e 1 X em 2 B t ' T em e 2 z x ^ '

Bv^^^ ^ B^\|f -, B^t 4 7.2 D—^= m (m P^ -^ + P'^ _J^) + (p4 _^ +U,^ P'i) ^ .n ,(4.4)

^t T L, e Z em ^ 3 em ^ ^2 L e z z ot ot jc3t 1 V ,^a where 11 , 11 , 11 are non-linear driving terms to be specified later,

y X z

'^

It is worth noting that a^ annihilates longitudinal magneto- ionic modes.

Furthermore

2

D = ^f„ (^f_„

^ + U"-^ ^? ) -^ + (JU,^ P^ P^

"^ " Bt^

em ^' em ' p ' "T ' e ' 2 ^ *L ' p

' e

' ^'^^ ^^

17
By (4. 5) relation (4. 4) can be given in the alternate forra

p

3^*^>

Qt

+ t . (4.6)

(2) (2) The second order polarizations, S and S , are related to each

other as follows

,(2) _ p em T e

uu P

,{2) _ ^L e

1

uu P^ P^

L p e P B/Bt em

s'/' = sf> (1

D

^ p2 p4

(jl) IS-' i)-' L p e

)

. (4.7)

Relation (4. 3) can now be transformed to

,(^) . s^'^ v^'^ + -^ = Si'> (v^^^

X

P d/dt

em

UU, P^

L e

).

(4.8)

(2) (2) Thus S - S , when D = 0, i. e. , when one has a

system (travelling wave) resonance. When this is the case

X 2 y

Now D can be written

2 2 P^D = P

P P

p {i+-4— + uu: ^_,^u<'''

3t

D

P" 9t

uu

,"]

P' -ft-^i'.^ (4.9)

D

X

18
The term D annihilates the natural ordinary magneto- ionic mode and

the plasma oscillation (for which f = 0), whereas D annihilates the

P X

natural extra- ordinary mode only. D can be used to study the

transient behavior of the magneto- ionic medium.

By (4. 9) one can also write

^,,2 D

2, ,2 Z, 22I4 = <{uj - iWq) (a; - (u^) - t^o '^T f ""o

V. / J 1 hz

where

k (o)) = — n (UU); and k (uJ) = — n (UJ) , Cq O X Cq X

and n and n denote ordinary and extraordinary refractive indices,

respectively. One notices that b = at the fourth reflection level

(referred to the angtilar frequency) at which | n | = °°. In the

neighborhood of this level, D is conveniently written

T^ o4 o2 ^ 2 I 1 1 \ I 1 D = - P P X , uu. 1-n l-n,;\i-n i-n,

o d' * X d

(X =UU^/UQ^), (4.12)

^ d d '

' where n , is the effective refractive index of the wave, of angular

frequency UU , upon which D operates.

Next, let us introduce the driving force polarization ratio'

2

^L d

^d = J^d = J;;^ 2- (4.13)

d 1 -X^ -n,

d d

19
Relation (4. 2) now yields

' d

(Y^ =UU^/UJ^). (4.14)

d

If the pxunp (high power) wave has the v velocity

VPP= V cos (UUPPt - k z); (k^PPPO = 0) n / c ) (4. 15)

where, quite arbitrarily, k could refer to an ordinary or extraordinary pumpwave, the other pumpwave velocities become

^p =

- Qp ^p ^i"^ ('"pt

- V'' ''p =mr"^"'V-V'*

(X =(JO^/uu^ (4.16)

with ^ ^

Q =Y, £-^ (Y^ =uu /O)). (4.17) ^ pl-X-n p

P P

The differential space charge density becomes

n i

pOc^l-Xp PP

The signal wave, of lingular frequency (W , is written:

s

i(uu t -k z) V j(uu t - k z)

''^ s s ' ^ s *'^ s s ' v=ve ;v=-Q-:— e ; S S S S J y X

''

"s = TTT- T

^

' Ps = pQ r- TTir- T

"

z s *' s ''

s

20
where

1 - n

Q = Y_

s L

s 1 - X - n

s s

s

(4. 20)

The terms f ,

i|/ , and t can be written [Rydbeck, 1962]: X y z 3w 3w s^ aw

" " " ° at >^ ^ >^ ° 9 t " at^ " ° at

(4. 21)

If we limit ourselves to a study of the sum and difference

frequency terms only, we find that

X Sv X.

p = p = " %

' ^ ^sd z a z

1 - X

21 - %

1 -

n. z a z 1 - n

(4. 22)

a V

p = p V

sd z a z

X \ s

n

av X. — V

z a z 1- n

(4. 23)

X. av 9v

P = P

z z z z V + V —

sd a z "s "p 1 - n \ a z y a z X

s ' X a V a V ^ V + V

l-nVaz y az x/

(4. 24)

and

PQ S p p S

(4. 25)

We evaluate next P , P , P , and W from these

X y z relations and obtain for the sum and difference terms (index + and - ,

respectively)

21
, ± -^ . s p 2

X

'^ d ±

U)

X X

<l--J)--s p^p a k Q 11- —2_|

2 1 + a

p

k

s Qs (lI , ^l>

2 + !

1-n/ \ 1-n

s

+ X — (a nQ +a nQ) ±Cq ssp pps

j(uu t-k z)

(4. 26)

V V

,± s p 2

^y 2 ±

U)

(1- -'> ^s^|i--^]*"p'^s (i-:^2i> +

+ X — (a n±a n)

± Cq s s p p

1 - n

j(U)^t-k^z)

1- n

1

(4. 27)

,± . s p 2

^z '' 2 ±

± a a {k^ + X^ — (n + n )} -

s ^p ± ± c^ s p^

1-n 1 - n ' ^

s p

j(aj^t-k^z)

(4. 28)

where

2, 2

k = k ± k ; X^ = UU^/uu^; a = Y^ /(I -X ); and a =Y^ /(I -X ).

± s p ± ± s T

We now write (4. 14) as follows:

p T )

(4. 29)

n=^ = ^f (l-Xj(l-X^-n^)|Q^ t%jt^+Y^Jl.n^)t^^[,

V V

S P F± (4. 30)

where

^±=% <"^d = ^±^' (4.31)

and by (4. 26), (4. 27), and (4. 28) obtain

Note: In what follows uu denotes U) , etc.

22
?± 2 (l-"f»^s^|l-7^l'<l*Q±

1 - n V*

X

± a k 1 - P s

1 - n / ±

± a a {k^ +X^ —(n + n )}

s p ± ±Cq s p

X k X k

1-

''z^Z^zl^'-^V

n 1 - n

s p

± a^ n (1 + Q Q )}

p p ± p

uu

JJ

p j(uu t - k z)

y + X^ — {a n (1 ± Q^ Q ) ± ' ± c s s ^ P

(4. 32)

First we notice that § =0, if (JU = 0. This is due to

the fact that the differential space charges and the longitudinal velocities

(which form longitudinal plasma wave pairs coupled to the transverse

wave by O) ) are zero in this case. Had we considered second order

terms in the pumpwave field (which could easily have been done) this

would not have been so. The transverse magnetic field thus is a very

important factor in the non-linear interaction theory.

When X -» 1, or X -1, we have longitudinal driving

force plasma resonance and | 5 | -^ » (in the collision-less case). In this

connection it is worth noting that

X

1 - n

X

Y Y —^ + 1 --^

(1 -X),

1-X Y^

„2

^ if X- 1 (Y^» |2(1-X)Y I) .

^ 1 + _ (1 - X) . '-\ ^T

If these driving force resonances occur in the travelling wave resonance

region, the generation of sum and difference frequency wave may be

23
greatly enhanced. Even if there is no resonance of the latter kind,

sudden driving force resonances may cause appreciable sum and

difference frequency radiation.

Another level of considerable interest is the fourth

reflection one, where n or n -» *. At this level, determined

s p

by b = (which indicates that it is gyro-plasma oscillation resonance

coupling), all electron velocities (and the differential space charge

density) become very large. This is in contrast to the plasma

resonance level, where only the longitudinal velocity becomes large

(and p if n = n ). s , p s , p x'

In ionospheric applications, this important resonance

level is difficult to reach, except from a topside sounder. Moreover,

travelling wave resonances, which are necessary, since D otherwise

woxild grow to large values (see (4. 10)), might be critical to obtain

at this level in practice, (b would have to be zero both for UU and

for uu or ou . ) s p

Before we continue the discussion, let us express the

velocities in the corresponding E -fields. One has

1- ^s e ^"^p ^ v = E , and V = ^ E , (4. 33)

s X mUJ s P X "^^r. Pv

s s y p P y

one finds that

2h
muu. X

E* = -j

i

^ V V

J (^ a ±

2

' "

1 j(u)^t-k^z)

± n a ) — e + v' ' p p c

(4. 34)

muu X

E = -J - ^

e 1,2- n

V V

s p (n a Q +

s s p

J

(uu t-k z) ^ , 1 ± ± (2) + naQ) — e +v^'

p p s' Cq X

We now obtain the final formulas,

^0 X.0 E E T

s p ±

E* ^-^

y 2 T T

s p

UU^(1-X ) (1-X - n^) 5"^/ d"^ -

n a ± n a j(ujt-kz)

s s p p

'' ± ±

+ —- e > .

E^=-

X

^0 ^0 E E T

s P ±

y y

T T

S p

(4.35)

(4. 36)

j Q^ UJ^(l-X^) (1-X^- n^) §="/ D^" ±

-0 r ± 3

X

0)^(1 -X^- n^)

(4. 37)

where

X m UU T =? P 1 - n

etc.

In order to study the stationary solutions, we present D

(4. 10) in an alternate form, viz. , 2 2 2 2

(n^- n^ ) (n^ - n^ ) D^ =X^(1-X )' ^ ^, (l-n^Xl-n^)

± ±

(4. 38)

25
where

n = k / uu ± ± ±

Since we want to study what happens at or near the

(4. 39)

travelling wave resonances (D =0), especially to obtain the

amplitudes of the natural sum and difference frequency waves, we

neglect (in this particular case) the insignificant last terms in (4. 36)

and (4. 37). By (4. 38) we can write

s p

y

2 2

UU UU {1 - n ) (1 - n ) p s p s m UU UU

(1- n^)(l-n2)

± ±

(n^ - n ) ( n^- n^ ) ± ±

1 - n

5*.

(1-X -n )

(4. 40)

E* = j q"^ E^ . (4. 40a) X y

Next, let us, in analogy with paragraph 2, determine the

natural waves that leave or approach the boxindary z = between the

ionized medium (medium 11) and vacuum (medixxm I).

We have the following sum and difference frequency waves

to consider:

Medium II j(^,t-k z) j(uu_^t-k z)

E =E%T<°) . '- ,T^-> e X X 0,±1 ^ 0,±1

E = E + y y jQ

j(uj t-k z)

^T«°> e ^ +-i

0.±1 jQ ^0,±1

j(uu t-k z)

T^^e

(4. 41)

26
where j Q and j Q are sum and difference frequency polarization

± ±

ratios of the natural sum and difference frequency waves.

Medium I

(1)

Jt'"±t + '"±^/Co) 1

E

(2) J(^t + u,^z/c^)

y o,±i

>

(4. 42)

If we break up the associated wave, E , in two

y

components, a and 3 , with the (natural) polarization ratios

Q and Q , we obtain

o X

± ±

a^ =- Q^/Q -1

± X

Q + 1 o±

Q^/Q -1

2 =" °±

°± Q^ + 1 o±

\

J

(4. 43)

The boundary conditions now yield

1- n.

• ± o

(x)

1- n

' ' = - B i Q

0,±1 ^±^ x^ 1 - n

±

X .

(4. 44)

n _ n

± o

n _ n

± X

n n

R

(2)

0,±1 ^± 1 -

+ 6 j Q -j

± x^ 1 - n

± X

n _ n

± 3C_L

± \

= a

n

+ e ± 1 - n

(4. 45)

27
The "reflected" field, i. e. , the wave leaving the boundary

(z = 0) in medium I, thus is composed of two parts, with polarizations

Q Oxand Q , respectively. Since the energy flow in the z-direction

± ±

^^ 2

p1'^= n (q'+1) Ie^I /Z.,

q q q y

the energy balance at the boundary requires that

Sign (n^ ) =

' ± d) n ± (JU n

> - Sign — ^ 2 P . (4 46)

s p

Sign ( n^ ) =

^ ±

For sum frequency waves n and n are thus always negative

(if we assume n to be positive, i. e. , a pumpwave incident upon the

It boundary in medixrm I). The natural sum frequency modes travel always

toward the boundary in medium II. When (i) < 0) , the natural

P s difference frequency modes travel towards the boundary, if k < k ,

and away from it, when k > k . When uu > UU the situation is

p s p s reversed.

If we make use of the fact that

and

Q±

- 1 = ^±

2

n

±

1 -

2

- n

X

±

Q i-x^.^l

2

n

X

±

Q:.

- 1 = X

±

2 2

- n

Q 1 - X - n^

± ±

1 - 2 ' n

(4. 47)

(4. 48)

28
the "reflected", frequency shifted field in mediiim I can finally be

written:

E = y

E E e ^^^^ (l-n)(l-n ) s p P s p s y lY _^

m UU (Jl) (JU ± ±

1+ n

1 - n

1 + n

Q +1 n + n

. o ± o

^ ± ±

'> ?* e

j UU (1 + n ) z/ c^

E = y

Q +1 n^ + n ^

E° E° e UU UU (l-n^(l-nf) f

j Q^ s p P s p s o

y y J ^

(4. 49)

1 + n

mUU UU^ UU 1 - n.

± ± ±

v^ ± ±

j Q. 1 + n

X j (W^ (1+ n^) z/Cq

Q +1 n + n

X. ± X

± ±J

(4. 50)

where it is to be remembered that Q Q = - 1. We notice from (4. 49) and (4. 50) that only two resonances

remain (compare the similar situation in paragraph 2), viz. ,

n ^ = - n , and n = - n , or

-(uU-±UU)n =0) n±UU n , s s P P

-(a)-±UU)n =CU n±UU n .

(4. 51)

s s P P

It is also important to note that these resonances by (4. 46) correspond

to the correct signs for n and n , and are thus physically

± ±

possible, if | n I = I (uu n ± UU n )/ (uu ± uu ) | , or

o s s p p s p

29
if In 1=1 (O) n ±aj n)/(uu ±uu)|. ' X s s p p s p

The possibilities to obtain these resonances are considerable, since n , n , n and n may vary within wide limits in

o^ x^ s P V

the magneto-ionic medium. The degenerate case UJ = 2 lU , which is

of particular interest if one wishes to study the possibilities of

obtaining second harmonic radiation or second harmonic echoes

(E = E = amplitude of high power primary wave, uu = UJ , etc. ; p s p s Jy y

? assumes a simpler form), has been studied extensively by the

author [Rydbeck, 1962].

The second degenerate case, OU = - UU (which leads to

the same resonance conditions, although § is different from § of

the previous case), is of particular interest, if one wishes to study the

possibilities of degenerate parametric amplification in the magnetoionic medium.

Finally, let us return to the influence of the resonances

of the driving forces, a situation that is somewhat different when we

(2) discuss the magnitude of E instead of v . y y

1. Fourth reflection resonance. Assvime that n -* ".

P

We then find from (4. 32) that

•± , 2,

. " (1 - n^) Hp .

1. e. , E

y

(medium I)

3

< 1+ n

1

1+ n

n + n

± o n + n

± x

±

30
which demonstrates that the influence of the driving force resonance

is extremely strong. This appears to be the most interesting nonlinear resonance in the magneto-ionic mediiim.

2. Plasma resonance .

X - 1; n<°^->0, n^^^^ 1. P P P

(x) a. n = n . One finds that

P P

2

1 - n r 1+ n 1 + n

s o± ^±

TT °^ . + y 2 '

. ,. , 1 - n n + n n + n (medium I) ±

' * °. ± X

±

In contrast to what might be expected, this driving force resonance

produces no enhanced effects.

b. n = n . One now finds that

P P

1+ n

+

1+ n

^±^^o^ n^ + n

!.

I^yl

^ ^l.n^ l-X

± p

i. e. , the effect of the driving force resonance is very profound. A

more detailed study of sum and difference frequency generation

caused by this resonance will appear in the final paragraph 6 of this

communication.

* * * A further analysis of the travelling wave resonances

(4. 51), interesting as it may be, is outside the scope and aim of the

present communication. That this interesting phenomena, applicable

to topside sounder ionogram interpretations, can be found seems

certain.

31
5. Generation of Natural Sum and Difference Frequency Waves

in a Slowly Varying Magneto -ionic Medium

In order to analyze the resonance radiation of sum and

difference frequency magneto- ionic modes, we make use of (4. 36),

which by (4. 10) for our present purposes can be written

"'— AC' ^

- +k —- +k ,2 o I \ , 2 X

^dz ±/ \dz : E — - y

^0 ^0 E E T

1 s p ± J. c^b

±

T T

s p

j(uu^t- / (k ± k )dz)

2 ± ±

± za s p

• (1-X^- n^) § = t e

' (5.1)

For the slowly varying medium the solution to (5. 1) now

can be written

+j /k dz 2 -j r(k ±k 4-k^ )d2

E' * * ' 1 / ^a ^-u r 2a s p X

'^k^ -k^ Ik

a '±

/

- j / k dz ^ - j ^ (k ± k - k ) dz

^a ^

- e • /

^a ^ P ^

dz

dz -

Z Z

J' dz -

- J y^k dz

z o a ±

- e /

j J^ (k ±k -k )d:

z^ s p o

a ^ dz (5. la)

32
If we assume that resonance takes place at some

specified level z , where, for example

k = - (k ± k ) , x^ s p'

and conditions are such (see paragraph 3) that a forward extra- ordinary

magneto-ionic mode is generated, we obtain (for definition of W and M

see paragraph 3)

z < z (W^ » 1)

j (O) t - / (k ± k ) dz)

±

± z^ s p

E «s ^ {k^^ -(k ±k )] [k^ -(k ±k )^} °± s p ^± s p

(5.2)

z > z (W » 1)

j (UU t- y (k ±k )dz)

E ?» (If e y

z^ s p

- j y (k ±k ) dz

[\.^ -(k ±k )^} [k^^ -(k ±k )^3

°± s p x_^ s p

+ i ( / k dz +n/4)

^0 ^±

(ko -kx )2 k

X

(5.3)

Although the amplitude coefficients are much more

complicated than in tiie isotropic case, dealt within paragraph 3, the

radiation feature is essentially the same. Furthermore, it should be

pointed out (see (4. 32) that the pvimpwave, for example, does not have

to be of extra- ordinary kind in ord.er to generate an extra-ordinary

33
natural mode. This is a typical feature of the non-linear propagation

theory.

In conclusion a few words should be said about the ray

paths. Since we are dealing with an anisotropic medium these will

deviate from the phase paths. In order to make the discussion more

general, we assume arbitrary incidence and noting the direct similarity

between the coupling integrals of (3. 5) and (5. 2), write the phase of the

downcoming difference frequency wave as follows:

%

= / (k -k -k )dz+ — (uo sin cp - uu sin Cp ) (x- - x,) , (5.4) ^ ^ Pz -^ ^0 ^ ^0 P Pq ^ ^ za

where x and x are the ray positions corresponding to z , i. e. ,

X for the downcoming difference frequency wave (note:

cp = (cp

) , etc.).

"O ' Nq =

If the signal wave carries the information, for example,

in pulses, the position of the downcoming difference frequency wave is

determined by the condition that

B0'

II

and by

= , (5. 5)

^ ^'

= , (5. 5a)

3cp

Po

if the pumpwave carries the information. The ray direction of the

z associated wave, exp. [j {uu t - (k ±k ) x - T (k ±k )dz } ], thus

± s p J s p X X

-^
has the ray direction of the signal wave or the piimpwave, if the former

or the latter carries the information. This is another typical feature

of the non-linear propagation theory.

/ 2 7z

Since n = J n - sin cp , etc. , and

s

'' s s z

1 sm cp = — (uu sm cp - o) sm cp ), we obtain

-

UU S S D D

0, -1 s - — = COS cp dcp

r'or

/

< tan cp + _

I s c 2 (

( 1 d n 1 dn

cos cp \ ^e ^ ^ ,

+ tan cp + dz +

cos cp \n d9

cp v^ ^

2k

-1 - ^^2-^1^ + U

\i (5.6)

For a very slowly varying medium, x - x thus

approximately becomes (compare (3. 14))

2k

^2"^i^ ro-\ +

1 \ z H ^

0^

(5.7)

Due to the "height dispersion, " the ray path of the downcoming difference

frequency wave is deformed, as if the frequency shift reflection took

.2k

-1

place at the level z +|

U

, instead of at z . For this change.

the wave normal paths in figure 3 have not been corrected.

35
Returning to our present case of magneto-ionic vertical

incidence (cp = = cp ), we obtain

S * X

. (5.7a)

The difference frequency wave therefore does not come down where the

1 d n 1 d n_

signal wave enters, if , q + ——5- + . n d y n d y

s Since

1 d n

' 2

= TY^ Y_ (1-n ) . ^ ^ ^

, (5.8)

X

^

e would have to be equal to 0° or 90° for \i safely to be equal to zero.

Since the non-linear driving terms disappear if 9 = 0° (unless we

consider second order quantities in the high power pump field), we

have to select = 90°, i. e. , transverse magneto-ionic propagation,

in order to obtain reliable travelling wave resonance effects recorded

at the same transmitting and receiving site (for example in a topside

sounder).

6. Sum and Difference Frequency Generation when the Non-Linear

Driving Forces Have a Resonance

For the sake of simplicity we limit ourselves to one

resonance only and assiome that the signal wave experiences longitudinal

plasma resonance at some specified level z = z . If H is the scale

36
height of the electron density, we write accordingly

We now have to include (a modest) collision frequency v , and replace

a^=Y^/(l-X^) by a^ = Y^ /(1-X^j6^), where S^ = v^/ ^3 • ^ ^ ^

(o)

If the signal wave is of ordinary. type, n = n (see

2, b, of paragraph 4), § near and at the resonance level can be

written

? - .^ (1 - n,^) S 1 \ ,1 ^ Q Q ) , —^ .

• (n^ + n^X^) e ^^

>, (X * 1) (6.2) ± p ± p I

where k = 0. We have (naturally) neglected triple split coupling of the

signal wave and likewise have assumed it to be of practically constant

amplitude through the important part of the resonance region, a helpful

but somewhat crude approximation.

Noticing that b can be written

b =uu^ (1-X ) X^ -^

—, (6.3)

(l-n^)(l-n'')

Relation (5. 1) yields

2 ,\/ 7 \ ,±

j{u,^t->(k ±k)dz3 d,2d 2± ik a ^

dz^ ±1 \dz^ ""i/ y 1-X -j6

37
where

2

uu uu IW 1 p s ±

f = .E^ -^

(1- n^)(l- n.^)(l- n^) (1-n^) (1-X^-nJ

• Y^ <|1-

X

1 - n

2I<1*°*V* 1 -X

n± + ^„X )> . (6.5)

and

V =E e/muu. , P„ P„ =^ (6.6)

i. e. , the ac electron velocity that the pximpwave would generate in an

isotropic medium.

Since the evaluation of ordinary and extra -ordinary

natural sum and difference frequency waves follows the same procedure,

we take the ordinary sxim and difference frequency waves as

characteristic examples. By (5. 1) we obtain approximately

^(o) 2 k^ - k^ k

O X o + + +

/

-j J^ (k +k +k ) dz

^a " P ° + dz

1 -X -j 5 s s (6.7)

-j/^ (k +k +k ) dj z. s p o/

H 3 e +

where it has to be remembered that k must be positive for a downo

^

+ coming wave. The coupling integral, 3 , now becomes (since k «i 0)

T S

38
y

+jy + / V - J a

where

3^ = 3 (y,(j) = ^-j^ dy, (6.8)

a = 6 spo, H (k +k ), and y = (z^Opo - z) (k +k ) (6. 9) + + One finds that

{j2ne , o>

jn , 0=0. (6. 10)

, a<

Assixming k to be positive (pumpwave running in positive z-direction),

ir we thus find that a siun frequency "echo" is obtained only if the electron

density gradient is positive. This is in accordance with the physical

situation, since the signal wave runs in the direction of increasing

X . s

In order to demonstrate how the "reflected" difference

frequency wave is being built up in the driving force resonance region,

we have in figure 4 presented the amplitude of the coupling integral as

a function of y. One notes the (infinite) plasma resonance for a =

and the moderate field perturbation in the resonance region when H < 0.

If we assume that the pump frequency is larger than

2uu , so that a difference frequency wave, in principle, could propagate

away from the resonance region, the coupling integral becomes

39
3 = 3(y,cr') = y + j

<^ dy,

where

a=6H(k s^p +k ), and y = (z^ - z) (k +k ), o'' ' ^0 p o'

It appears from (6. 10) that

o

-jZTTe , o>

3( + oo, o) = { - j T] , a =

, a<

(6. 11)

(6.12)

(6. 13)

Also in the difference frequency case (u) > Zuu ) a positive electron

density gradient is required for a downcoming wave to be generated.

For the ordinary forward sum frequency wave, the

coupling integral becomes

' a= 6 H(k -k ) s p o^

3 = 3(y, a), where (k > k ) P o +

* 3=3 (y, a), where <

y = (z-z^) (k -k ) ^ ^ O' ^ p o^'

a= 6 H(k -k ) s o_^ p

(6. 14)

(k < k ) P

o

y = (z -z^) (k -k ) ^ ^ O'^ o^ p'

Thus, even the generation of a forward wave requires a positive

electron density gradient.

The downcoming suan frequency wave, at the level

z = z , finally can be written

IK)
(6. 15)

where

R

(o)

0, +1

-6 H k +k

s ' p o. ,nk H e

Z c o , \ + P

z = z

j(uut -/(k +k +k ) dz)

T -7. ° P U I +

2 2 2 2

rUUiA) (l-rL)(l-n ){l-n ) p^ s p^ o X

+ +

2 +

2 2 2

( n - n ) n

o X o + + + X

s 1 - n

^ P

2((l-^V

(1 -X - n ) :

( n +X n )

^ + + p'

1 -X -> z = z

Since

^s ^

0) p= p

+ for H = max

V e k +k

s ' p o V |k +k I

(6. 16)

(6. 17)

p o +

" + an optimum that should be attainable at some levels in the ionosphere.

(o)

the order of magnitude of I Rl , I is v ou /c^v , -if uu is ^ ' 0, +1

p c s' p

moderately larger than 2 ou .

If we assume the pumpwave (the high power wave) to be

pulsed, and the signal wave to be continuous, the s\am frequency "echo"

delay becomes

kl
^0

^°'

/ (k +k +k ) dz= — {z^ -z +H C -+ 0}. (6- 18)

P z.

CL

if uu is so large that (n ) =1. For the difference frequency "echo"

delay we similarly obtain

, » 2 2 uu

provided that uu - 0) is so great, that (n ) = 1. These relations

p s o z

- yield

H=^ (,(°).,(°)), (6.20)

and

c« . . , . . » . » 2 u)

r o) (o) , (o) (o), p .

s By sum and difference frequency pulse "echo" measurement of this kind

it is thus in principle possible (if (i) - (i) is large enough) to obtain

P s not only the true ordinary reflection height, z (X = 1), but also the

u s electron density gradient, N /H, at the same. Since the pulse

measurements have to be made at relatively high frequencies, for

example f = 30 Mc/s, t and t can be determiined with much

S T ~ greater accuracy than at the unshifted signal frequency U) . Another

advantage is the fact that uu , if it is large enough, can remiain

r unchanged, while ou is swept through the probing frequency range of

s interest.

h2
The immediate question one asks oneself is the following:

is it practically possible to perform sum and difference frequencypulse echo measurements of the type just discussed, using ground2 2 (o) based equipment? If UU ^^ iU , the amplitude of R-^ , by (6. 16)

approximately becomies

4

-p -sn

K.+ii -iT^~r"^ "":—j~ ""t ' (6-21) ^'

'^O S t .^ (UU +11) ) UU s ^''^ ' p s p

where

V

u = — H Ik +k 1. (6. 22)

ou

' p o

' ^ ' s ^ +

If we assume u to lie in the range 0. 08 to 4, i. e. ,

within a u-ratio of 50, uS: o. 20 (Li = 1), and U) /2n = 30 Mc/s, max p

(1) /2tt = 5 Mc/s, we obtain r> 2 V f

'^O.+l' c, V ^° ^T •

* s s V

3 o ^v which with V =10 yields |r^ , I ^ —^ Y^ If the ground-based

s pump transmitter has an effective radiation aperture A, and the

ionospheric absorption is neglected at the pump frequency, v

becomes

where P is the transmitter power and r the distance from the

ground to the z -level. Considering the fact that the returning sum

hZ
(or difference) frequency travels practically without absorption and

that the atmospheric noise level at the sum (or difference) frequency

is very low, compared to at f =5 Mc/s, -we may assume that a

-5

V /c„ -ratio of 10 would yield a detectable echo at the ground

p

' ^ y

(provided the regular iono sonde at 5 Mc/s is powerful enough). With

4 2

an effective antenna area of 10 m , and r = 300 km, P

becomes 5 Mw. It thus seenns to lie within the possibilities of present

day techniques to perform ionospheric sum and difference frequency

measurements as outlined. Beyond doubt practical experiments of

this type would be very interesting and probably also rewarding.

Acknowledgements

The research reported in this communication was

partly done at the Ionosphere Research Laboratory (IRL), Pennsylvania

State University, and partly at the Research Laboratory of Electronics,

Gothenburg, Sweden. The author is much indebted to the IRL and to

the U. S. Air Force, for support under Contract AF 6l(052)-451.

7. References

Rydbeck, O. E. H. (1962), Dynamic non-linear wave propagation in

ionized media, Part 2, The Magneto -Ionic Medium, Res. Rept.

No. 28, Research Laboratory of Electronics, Chalmers

University of Technology, Gothenburg, Sweden.

Rydbeck, O. E. H. (1963), Electromagnetic non-linear interaction and

reflection from a plane ionized medium, Sci. Rept. No. 183,

IRL, Pennsylvania State University.

kk
M«41ua II —

/

2 -'\

/

\\\\\\\\\\\\\iA\\ wwwwww /

y(\\\\\\\\\\\\\ ^ z<0 1

L

X

Meditun I

/<±p «o- 0-

H

-<Pi

l-£,

-H IS z

Hiffh pov«r Slsnal waT«,

*VuBp" vaT«, angular fraquaaoy o) ani^ular fraquenoy oi

Figure 1. Depicting polarizations and angles of incidence of signal

and pijmp waves.

k3
H

/

/

t

\/

T \ -H

N \^ -^ —?*

/ T~

/ ®

/ / J-« / / t

/

/

/

/

/

\\V\V /

/

/ \\h\ / 4 If / 6 3

^^ •H / o U •

/ P« /

II 4. S

/ a

/ H • H

/ e- •0 *

y O &

H

ii6
g

bO 1

o

QJ p

•H

CO>sO

aI

I

o

H

O

id

0)I

w

a

•HI

i

oo
hQ
Non-Linear Interaction Coefficients for Electrons

in Nitrogen and Air*

A. V. Phelps

Westinghouse Research Laboratories

Pittsburgh, Pennsylvania

An analysis of measured electron transport coefficients for

nitrogen and oxygen has yielded a consistent set of energy dependent cross

sections for the momentum transfer and inelastic scattering of electrons in

these gases. These results are used with the appropriate Boltzmann transport

equation to predict the effects of the heating of electrons in nitrogen and air

by radio frequency fields. The rapid increase in mean energy with electric

field predicted by Altshuler at high frequencies is obtained, but the calculations do not show the predicted hysteresis effect. These procedures have

made possible the direct calculation of cross modulation coefficients rather

than the usual procedure of first evaluating the change of mean energy produced

by an electric field and then calculating the change in propagation constant.

if This work was supported in part by the Air Force Weapons Laboratory.

h9
I. Introduction

The purpose of this paper is to present the results of improved

calculations of non-linear interaction coefficients for electrons in nitrogen

and in air. In the past the lack of accurate electron collision cross section

data and of accurate procedures for determining electron energy distribution

functions has made it necessary to calculate the effect of electric fields on

the electron transport coefficients and reaction rates under simplifying assumptions regarding the electron energy distribution and the electron energy

loss processes. Recent studies have considerably improved our theoretical

understanding of electron collision cross sections at low energies £^\, 2j7

and have led to the determination of reasonably accurate and realistic electron

collision cross sections in nitrogen ^2),Aj^ and to the approximate collision

cross sections for electrons in oxygen given in this paper. When these cross

sections are used with an accurate form of the Boltzmann transport equation to

determine the electron energy distribution function and various electron

transport coefficients we expect to obtain accurate values for the various

Italicized figures in brackets indicate the literature reference

at the end of this paper.

50
coefficients describing the non-linear interaction of electromagnetic waves

in weakly ionized nitrogen and air. The collision cross section data and

the appropriate Boltzmann equation are summarized in Section II. In

Section III, the variation of the mean electron energy with electric field

strength obtained from our calculations is compared with the results of various

approximate calculations . In Section IV, a comparison is made of experimental

and theoretical determinations of the coefficient of cross modulation for waves

of nearly equal frequency

II. Cross Sections and Boltzmann Equation

The cross sections for momentum transfer and inelastic collisions

for electrons in nitrogen and oxygen which are used in our analysis are shown

in Figures 1 and 2. The cross sections shown in Figure 1 for nitrogen have

been derived to be consistent with dc measurements of electron drift velocity,

of the ratio of the electron diffusion coefficient to electron mobility, and of

excitation and ionization coefficients Z 4_7. As has been pointed out

previously /"3_y, although the cross sections derived by this technique are

not unique they are much more accurate than previously available data and are

consistent with most of the known experimental data. These effective cross

sections are believed to be accurate to + 10% or better. The cross sections

shown for oxygen were determined by the same techniques as used for H^ ^^^

51
and N /"s, 4_J . These results are tentative largely because of the lack

of consistent experimental data. Our primary concern is with the cross

section, O , , for vibrational excitation of 0„ which has been neglected

vib ^

in most previous calculations in spite of the fact that it is the dominant

energy loss mechanism in air for energies between 0.2 and 1.7 eV. From

the agreement between calculated and measured values of the drift velocity

and the ratio of the diffusion coefficient to the mobility, we believe that the

effective cross section for vibrational excitation is accurate to + 30% in this

energy range. The momentum transfer cross section is believed to be accurate

to + 20% in the energy range from .2 to 20 eV. Outside these energy ranges

the momentum transfer and excitation cross sections shown are much less

certain but are the best available at present. In particular, we have not shown

rotational excitation cross sections for since theory /Tsj^ indicates cross

sections which are an order of magnitude smaller than for N„.

The Boltzmann equation and the integrals for the evaluation of

the electron mobility applicable to the calculation of the behavior of

electrons in weakly ionized N and air are given by Equations (1) through

(7) of an earlier paper /~7j7. The mean energy of the electrons is given by

I = fi^^^ fit) d£. (1) -'o

52
where £ is the electron energy and f(£) is the normalized electron energy

distribution function /~7_7.

Ill . Mean Electron Energy

Most previous calculations of non-linear effects have made use

of approximate analytical forms for the electron energy distribution function,

e.g., a Maxwellian energy distribution. As a result, it has been convenient

to calculate the changes in the mean electron energy resulting from the

disturbing electromagnetic field and then calculate the corresponding changes

in observable quantities such as the attenuation coefficient for an electromagnetic wave. In the present discussion we will be concerned with the

rather large changes in mean electron energy such as those considered in

calculations which lead to a prediction of a "hysteresis effect" in the curve

of mean energy as a function of electric field, E, at high frequencies f^J

In Altshuler's investigation expressions were derived for the mean electron

energy which are based on the assumptions that a) the electrons lose energy to

nitrogen molecules only by rotational excitation, b) the effective electron collision frequency is a linear function of the mean electron energy and c) the

electron energy distribution function is Maxwellian. Since these expressions

were not expected to be valid at the higher mean energies where other inelastic

collision processes are important, we have obtained progressively more accurate

53
values for the mean energy by relaxing assumptions (a), (b)jand (c)

One simple way of including the effect of vibrational excitation

is to equate Altshuler's expression /"sj^ for the power input per electron

in the radio frequency case to the power input per electron at the same

average electron energy intiie dc limit and solve for the ac field required

to maintain the electrons of this mean energy. If assumptions (b) and (c)

are correct then this procedure should yield the same result as a more accurate

analytical or machine calculation. Thus,

P t e E e E

- ^

(2)

m(a) + (3 ^ ) mp g: where e is the electronic charge, m is the electron mass, E and E, are dc

the rms ac and the dc electric fields required to maintain the mean electron

energy at j; , and 3 i: i s the effective value of the frequency of momentum

transfer collisions. In the high frequency limit Eq. (2) reduces to

0) P £ (P/N) E" (3)

Since no measurements of mean electron energy as a function of E /N are

available^ we have used the results of calculations jT^J ' The value of the

The author does not know of any technique for the measurement of

mean electron energy in gases where the electron energy distribution

may depart significantly from Maxwellian.

5^
coefficient (3 in Eq. (3) is taken to be the mean of the effective collision

frequencies in the low frequency and high frequency limits C^y. The

results of the application of Eq. (3) to N at 11 K are shown by the short

dashed curve of Figure 3. This curve shows that calculated values of £

are a multivalued function of the electric field parameter, E/w, at both low

energies where rotational excitation is the dominant energy loss and at high

energies where vibrational and electronic excitation are dominant. We note

that the decrease in E/w with increasing £. is only about 10% and that if the

calculations had been made at significantly higher gas temperatures no

decrease in E/w would have been obtained.

Our next step in improving the accuracy of the calculation is to

take into account the fact that at the higher electron energies the electron

collision frequency is considerably smaller than that assumed in Eqs . (2) and

(3) . A simple approximate way to take this into account is to replace the

expression for the effective collision frequency in terms of the mean energy

by the effective frequency of momentum transfer collisions, iJ /N, determined

directly from measurements of the electron drift velocity at a given Edc/N Z"3, 7^.

Thus, we use the relation

u (i; /N) m

(4)

55
to calculate the curve of £ vs E/w shown by the short-dash, long-dash

curve of Figure 3 . We see that the results are essentially unchanged at

low energies where the electron collision frequency is approximately a

linear function of the electron energy but that at large mean energies the

values of E/go required to maintain a given mean energy are considerably

larger than obtained with Eq. (3) .

Finally we can relax the assumption that the electron energy

distribution is independent of frequency for a given mean energy by

calculating the distribution function using the Boltzmann equation appropriate to the ac electric field £l^ and the cross sections shown in

Figure 1. The resultant values of £ as a function of E/w, as indicated by

the solid line of Figure 3, show that although the values of E/w required to

produce a given value of £ are not much different from those predicted by

the approximate calculations, the values of £ are now a single valued function

of E/(jj . The explanation for the differert character of the £. vs E/w curves is

that the shape of the electron energy distribution function varies considerably

with frequency . This is shown in Fig . 4 where the calculated electron energy

1/2 —18 2 distribution, £ f(£), are shown for a dc case, E/N=l,2 x 10 V~cm , and

-10 -7 3 a very high frequency ac case, E/oj = 2 x 10 V-cm and w/N 6 x 10 cm /sec.

In both cases the mean electron energy is 0.067 eV. Also shown is a Maxwellian

electron energy distribution having the same mean energy. Of particular interest

56
is the fact that, although the energy distribution in the do case is intermediate between the Maxwellian and Druyvesteyn forms /I'iJ, the

distribution in the ac case peaks at very low energies and has a very long

"tail" at high energies. Since the energy loss processes are the same in

both cases, the differences are due to the energy dependence of the energy

gain term /"8_7. At the highest & of Figure 3 the electron energy distribution

functions for a given mean energy are much more nearly independent of the

frequency

Figure 3 also shows a curve of mean energy as a function of E/w

for air at 232 K as calculated using the cross section data of Figures 1 and 2.

Here we see a much more uniform rise in £ with E/co which is due in part to

the higher gas temperature and in part to the large energy electron energy

losses in vibrational excitation of the oxygen at low £ .

IV. Cross Modulation Coefficients

In this section we are concerned with the accuracy of cross

modulation coefficients calculated using various approximations to the

true electron behavior in air. Throughout this discussion we will assume

that a) the electric field strength present in the gas produces an arbitrarily

small perturbatiori of the electrons from their thermal equilibrium conditions,

b) the electric field occurs in the form of a pulse whose duration is long

compared to energy relaxation times but short compared to the time required

57
for significant changes in the electron density, c) that magnetic field

effects may be neglected, and d) the effects of electron collisions with

oxygen molecules, with minor constituents such as water vapor, and with

other electrons, ions or excited species can be neglected. As a result of

assumption (d) the density, N, referred to is the nitrogen density. If

further investigation of the properties of electrons in oxygen supports the

tentative curves for Q below 0.2eV shown in Fig. 2, then the effects of

m

oxygen will have to be included in any more accurate analysis

It is convenient to use as a basis for our discussion of cross

modulation effects the theory developed by Barrington and Thrane Z~1Q_7 and

discussed more recently by Rumi ^11_7« This theory is based on the assumption that the collision frequency for momentum transfer collisions is a

linear function of energy ^5^» that the electron energy distribution is accurately represented by a Maxwellian distribution, and that the attenuation

per wavelength is small . The theoretical results for the fractional modulation

SE/E of a wave at an angular frequency w resulting from electron heating by

a wave at co. can be written in the following form

2 2 2 S E _ cop ^h e E (0. R , (5)

E w^ -^ 2m u^T N

where R

= '^ -^ ^^ ^

a\ ^I'^z'

^

^'"l*'

3 GkT w e 2

(6)

58
^2 = V^2' ''l = V'^2'

Va)=15|^C3/2^a)-ZC^/3(a)] . (7)

and f(a) = 5 ^c/o^^'^ •

2 2 Here co s? ne /m£ is the plasma resonance frequency where n is the

P o

electron density and & is the permittivity of free space; v* is the electron o I collision frequency for electrons with an energy of kTas defined in reference 9; ^h is the length of path over which the absorption occurs; A is the free space

wavelength; k is Boltzmann's constant; T is the gas temperature; G is the

effective value of the energy loss per momentum transfer collision; N is the

nitrogen density; and CJy (a) is a function defined and tabulated by Dingle,

Arndt and Roy /I2_J . The particular form of the function R in Eq . (6) was

chosen because it is approximately symmetrical with respect to frequencies

above and below the electron collision frequency and because it yields a

function which is roughly independent of temperature. This function is shown

by the solid curves of Figure 5 and 6 for the case in which u>^ = w , i.e., for

the case in which the wanted wave and the disturbing wave have essentially

the same frequency, as in the case of many laboratory experiments or in self

modulation experiments . The curves of Figures 5 and 6 were calculated for

l/'g/N = 1 .2 x 10~^(kT/e)cmVsec and for G = 3 x 10""^(300/T) . The value of

59
GT was chosen to fit the results of laboratory dc measurements at temperatures of interest in the ionosphere. For this assumption regardingthe

temperature dependence of G the solid curves m Figs . 5 and 6 are independent

of temperature

The only high frequency laboratory measurements known to the

author in which the assumptions listed above are satisfied are those of

Phelps, Fundingsland and Brown /"iS^. These results are shown by the

points in Figure 6 . Unfortunately these authors have not estimated the accuracy

of their electric field measurements. The ionospheric measurements analyzed

by Barrington and Thrane^lO^ using this theor/ indicated a G value of about

half the value we have adopted in order to obtain a fit to the dc measurements

A reduced value of G such as this would give a better fit to the experimental

points in Figure 6 but would require that the true R function deviate significantly from the curves shown. The obvious next step in this investigation is

to solve the Boltzmann equation appropriate to near thermal conditions at high

frequencies as has been done for the dc case in literature references 3 and 4.

Calculations by the author for gases in which the only energy losses are due

to elastic recoil show that it is not proper to assume that the perturbed electron

energy distribution function is that of Maxwellian distribution at a slightly

higher temperature

60
V. Summary

The calculations of the dependence of the mean energy on the

electric field show that one must take into account the changes in the

form of the electron energy distribution which occur as the frequency of

the applied signal is varied . Similarly, a comparison of the results of

a simple theoretical evaluation of the cross modulation coefficients for N

and air suggests that it is not satisfactory to assume that the electron

energy distribution function is Maxwellian.

The author wishes to acknowledge helpful discussions of this

problem with his associates in the Atomic Physics Group, especially

G. J. Schulz and A. G. Engelhardt.

61
References

1) E. Gerjuoy and S. Stein, Phys . Rev. 9Z, 1671 (1955) and 98. 1848 (1955).

2) A. Dalgarno and W. Moffett, Indian Academy of Sciences Symposium on

Collision Processes, 1962 (unpublished).

3) L. S. Frost and A. V. Phelps, Phys. Rev. 127 . 1621 (1962).

4) A. G. Engelhardt, A. V. Phelps and C. G. Risk, Paper E-2, 16th Gaseous

Electronics Conference, Pittsburgh, Pa., October, 1963 and Bull. Am.

Phys. Soc, Series II, 1, (1964).

5) A. Dalgarno, Ann. Geophys . 17. 16 (1961).

7) A. G. Engelhardt and A. V. Phelps, Phys. Rev. 131 . 2115 (1963).

8) S. Altshuler, Jour. Geophys. Res. 68. 4707 (1963).

9) A. V. Phelps, Jour. Appl. Phys. 3i, 1723 (1960).

10) R. E. Harrington and E. Thrane, J. Atmos . and Terrest. Phys. 24,

31 (1962).

11) G. C. Rumi, I.R.E. Trans, on Antennas and Propagation AP-10, 594 (1962).

Our numerical values for f. (o) differ significantly from those given by Rumi

for Q>10.

12) R. B. Dingle, D. Arndt and S . K. Roy, Appl. Sci. Res. 6B. 155 (1956).

13) A. V. Phelps, O. T. Fundingsland and S. C. Brown, Phys. Rev. 84.

559 (1951).

62
CURVE 568825-A

Momentum Transfer and Excitation Cross Sections for Electrons in N,

Fig. 1.

I

10 10 10

^ 1 Electron Energy (eV)

Electron collision cross sections in No for momentum transfer Q , rotational ekcitation and deexcitation q , vibrational excitation

Q .,, electronic excitation Q , and ionization Q,. For clarity only

one pair of rotational excitation and deexcitation curves is shown,

i.e., Q,^ for excitation from the J = 4 level and Q/-, for deexcitation

from the J = 6 level to the J = 4 level. Similarly, only the sums of

the vibrational and electronic excitation cross sections are shown.

63
CURVE 568826-A

Electron Collision Cross Sections in Oxygen

10

-14

10

-15

CVJe

10-16

c

o

O

10

-18

10

-19

-\ I \1 1 1 1 1 II II 1 1 II 1 1 I+- — -^ "m-^ — , ^ ^

—

ion -y^

— ^

-^i

:

—

vib 1

A

V

—

^att-j \

1 1 1 1 1 1 1 1

1 1 1 1

1 1 1 tI

10

-3 -2 -1

10

^

10

^

1 Electron Energy (eV)10 10'

Fig. 2. Electron collision cross sections in Op for momentum transfer

Qm, vibrational excitation Q ., , dissociative attachment Q , vib' ^att'

electronic excitation Q -, and ionization Q,. Only the sum of

the various electronic cross sections is shown on this graph.

0^-
CURVE 568828-A

Mean Electron Energy vs. E/u for Nitrogen

10

>

ii -1

fe 10

^

UJ

c

ov_ "o

03c 10

10

-3

10

-11

- 1 1 1

1 1

1 1

1

1 / 1 1 1 ^ 1 M 11 1 14-

—

if)

1 '/ ?

^"^— Air - 232*»K

—

— -^

k

—

_^

^^- N2-77«>K — —

1 1 1

1

1 1 1 till 1 1 1 1 1 1 1 1 1

10

-10 r9

10 10

E/u (V- sec/cm)

-8

10

-7

10

-6

Fig. 3 Mean electron energy of electrons in Np at 770K and air at 232°K

as a function of e/oj at high frequencies. The calculations were

-7 3 "9 -l

made for a/^ = 6 x 10 cm /sec for E/oj < 3.33 x 10 V-cm -sec and for 6 x lO" cm"^/sec for e/cj < 3.33 x 10 v/cm' -sec.

65
CURVE 568824-A

AC and DC Electron Energy Distribution Functions for N^

I

I 5 10

CM

i 1

o

c

Z3

LU

•- -1 3 10

^ X3

«/)

g" 10

2

c

o

.^ 10 ^

- 1 1 1 1 1 1 —

— e • 0. 067 eV —

k\

- —

— \ ^\ E o ,n-10

—

— ~^^ u/2xl0^" —

— )l^

^

V- cm" 'sec — \\\\^^^..^^-

— \ \

^

, J

\ \ /Maxwellian

^^

lO'

-18^

cm

—

__ V- \ \ ,._ \ \

1 1 ,\, \ , 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Electron Energy (eV)

Fig. 4. Electron energy distribution functions for

electrons in Np at 77°K.

66
CURVE 568795-A

Modulation Coefficient for Nitrogen

3.0 xlO

10

I'l l—I—I I 11—I—I I 11—I

I I 1

1—I—

2.0

1.0

E

^

-1.0

-2.0

-3.0 I \ I I I I L

I I I I I I I I I I

I I I I

10 10^ 10' 10' 10

8

(?) (?) - <eV-sec-cm-3,

Fig. 5. .Cross modulation function R.

10'

67
CURVE 568827-A

10

11

1 TT

Modulation Coefficient for Nitrogen

TT \ TTT] 1 TTTT

E

5"

10

10

10'

10' J LZU

10 10-

1—rrn—1—rrn

GOxlO-^(^) ~_

I 303 *»K

A 422 ^'K • 511 *»K

LL

10'

373 '^K

300 ''K:

u.

(r)(f)-'«v—-^

Fig. 6. Absolute magnitude of cross modulation function R

showing theoretical curve and experimental data.

10'

68
The Excitation of Optical Radiation "by High Power Density Eadio Beams

Lawrence R. Megill

Central Radio Propagation Laboratory, National Bureau of Standards,

Boulder, Colorado

We calc\ilate the power density required to excite various states

of 0„, and which may he expected to radiate light observable from the

groTind. These calculations are made by first calculating the electrons

energy distribution functions in the presence of the rf electric fields,

and from this the rate of excitation of the radiating states. We discuss

the feasibility of experiments to generate sufficient intensity to

cause appreciable modification in the night airglow intensity. The

possibility of performing experiments using the two body dissociative

attachment of electrons is also considered, especially in regard to the

performance of basic experiments in atomic physics.

69
1. Introduction

In the discussion of non-linear processes in the ionosphere, the

excitation of optical radiation hy a radio 'beam is an extreme case. Yet,

if transmitters sufficiently powerfvil to excite optical radiation in the

ionosphere existed, a great deal of quantitative information concerning

the constitution of the ionosphere could he collected on a routine hasis.

It is, therefore, of some importance to investigate the conditions under

which detectahle optical radiation might he excited.

This prohlem has heen considered "by others, Bailey [1959]^ Clavier

[I96I]; but not in detail which we will apply. Authors in the past have

estimated the intensity to he expected in the atmosphere hased on laboratory data obtained in a discharge. Such a procedure is liable to considerable error, since conditions in a discharge differ greatly from those

extant in the atmosphere. In particular, in discharges there exist many

metastable states which change the power required to heat electrons and

recombination processes which yield a great deal of the luminosity. In

addition, the electron density in discharges is often such that the electron

energy distribution is dominated by electron interactions, a condition

which does not apply in the lower ionosphere. [Megill and Carleton, 1964].

We shall, in this paper, concern ourselves principally with the direct

"heating" of the electrons already in existence in the ionosphere by means

of electric fields and with the excitation of electronic states in the

neutral gas by collisions with these electrons. The problem of "breaking

down" the atmosphere will be discussed only briefly.

TO
2. Excitation of Optical Radiation

If we know the distribution f (u) of electrons with energy, u, \uider

the influence of electric and magnetic fields then we can calculate the

total rate of excitation of a particular state V. . frcm

00

V. . = N. N / a. . V f (u) u2 du; (l)

where N. is the nvunber density of the gas, N is the number density of

electrons, a; . is the cross section for excitation to state j, v is the

electron velocity, and we have normalized the distrihution function so that

1 f(u) u2 du = 1. (2) /

Carleton and Megill [1962] have descrihed the calculation of the electron

energy distrihution function for altitudes of interest here. It is shown

that the distrihution function may he determined hy solving the following

differential equation.

[KT 6 v(u) + Y a v(u)] || = - 6 v(u) f(u) - Q(u) ^3)

where

k is the Boltzmann constant

T is the temperature of the neutral particles

V is the elastic collision frequency 2 u. a. v where N. is the density

of gas i and a is the elastic scattering cross section for momentum transfer.

4 is the average fractional energy lost per elastic collision -rr where

m is the electron mass and M is the mass of the gas pajrbicle.

71
2

y ±s rr- ( —) J e "being the electronic charge and E the rms electric

field, „ T - / - /

Q(u) is y N^ u ^ (m/2) ^x

00

/"[u' + u,,) a (u' + u..) f (u' + u ) - u' a., (u' ) f (u')] du'

where a; . and N. are as in equation (l) and u. . is the energy required to

excite the state u. .. The summation is over all inelastic states j for

each gas i.

a is a scaling parameter on the electric field expressed as

2 2 2

B

[v2 + (CD . u) )2][^2 ^ (^ ^

^2

B' ^"^ ^"^ B'

where co is the angular frequency of the exciting field E and u) is the gyro

B

eB

frequency of the electron in the earth's magnetic field B where '^n = — •

The solution of (3) is carried out numerically in order to ohtain the

desired distribution function, using experimental cross sections where available, and theoretically estimated ones elsewhere. A list of the cross sections

used for these and other calciilations has been given by Megill and Carleton

3. Scaling Parameters

While one may calculate the expected excitation for any given power

density, excitation frequency, and collision frequency, the number of such

calculations required to cover all possible cases is very large. We search,

therefore, for scaling parameters in the theory which will allow more

general application of calciilated values.

%

72
If we investigate (3) in detail, we find that (aftier dividing through

T^y ^'(tJ-) fo^ convenience) the left hand side of the equation constitutes a

kind of "driving function" which describes the manner in which the electrons

gain energy while the right hand side describes the mechanisms which will

cause the electrons to lose energy. We further see that on the right hand

side of (3) only the parameters of the gas enter. The only reference to the

mechanism which increases the electron energy then, is inside the "brackets

on the left hand side of the equation (kT 6 + Y a) . Recall that we have

divided the equation by v(u)

The term kT^ presents a base energy (due to the neutral gas temperature) which is the electron energy for zero electric field. We have, then,

as the driving function e by which the electrons gain energy

2 2 2

e = oM—) —^ -p p p-- (^)

3 "^ [v^ + («) - ^^fllv^ + (CO + (.g)^]

Now let us exatEine expression (4) for limiting cases. In the ionosphere we

have a rapidly decreasing nxaaber density with height so that the collision

frequency v = N a v is also a rapidly varying function of height. Consider

a beam of radiation extending upward through the ionosphere. It will,

except for a small transition region, be in a situation in which v » cju ± uig

or V « a>p« -An exception is the condition cu = (« about which more will be

said later. We will, accordingly, consider these two cases in sequence.

When V « u) and if uig < co^ 2 e^ ,Ex2

' = 3~y

73
"Which indicates that the mechanism hy which the electrons gain energy is

independent of the pressiore of the gas and is only dependent upon its

character as described by the right hand side of (s)*

On the other hand, if v » U) :t co-p then {k) reduces to

2 e^ /En2

^ =

3 S" ( v)

which, since v "^ N and N is proportional to the pressiire, yields the scaling

factor e/p which is commonly used in gaseous electronics.

When CD = uj-dj gy^o resonance exists. For purposes of calculating distrihution functions, this may "be considered to he the same as if there were a

dc field. This is strictly true only if the radiation is circularly polarized

in the extraordinary mode. Jbr most purposes of interest in ionospheric

problems, however, such an approximation is adequate.

In general, the electric field of importance in these calculations

will he that present in the ionosphere at the height for which the calculations

are heing made. This field is, of coiorse, dependent upon the power of the

transmitter, the gain of the antenna and the absorption of the radiation

below the point of interest. The calculation of the absorption of the radiation

is in itself a difficult problem which is not discussed here.

We shall define two parameters then, in terms of which we shall present

the data. These we will call the hf or high frequency and the If or low

frequency reduced power (P ) eind (P )_ where

(^rV =

Tf (|)' (5a)

and

(^r>l^ ^ ^

(!'• (51=)

Ih
If E is in volts per cm and T] is in ohms then these qioantities have

—P P —P ^ h. Tonits of watts cm sec and watts cm cm or watts cm if N is in units of

nuiriber per cubic centimeter. These imits, while they may appear \anwieldly,

are convenient for application to geophysical problems.

The power required from an antenna in order to generate a specific

effect may in twcn "be calculated from

P = ,7

^

h^ . / (6a) antenna AG ^ ' or ^4. = ,r ^ ' N (6b) antenna AG ^ '

where h is the height at which the excitation rate is required, N is the

numher density at that height and G is the gain of the antenna. A is the

amount of power ahsorhed "below height h. For the purposes of our further

calculations A is considered to he one.

4. Ohservahle Radiations

Before calculating excitation rates of various states, let us first consider what radiations are likely to he visihle from ground level. The first

condition, of course, is that the experiment he done at night. Because we are

dealing here with excitation hy low energy electrons which already exist in the

ionosphere as opposed to those created in a discharge we need consider radiation

from states which can be excited from ground state atoms or molecules. The

density of metastable states which can he excited to higher levels and

will then radiate are negligible. GroTond state rotational or vibrational

states of the major molecular constituents will not radiate, and so, energy

75
put into these states will not result in visilDle radiation. We discuss here

only the excitation of electronic levels in the atoms and molecioles which may

possihly re -radiate. In calcTilation of the distrihution fimction, however,

all inelastic states have "been considered even though they are not discussed

explicitly in this paper. In nitrogen, such states require so much energy

to "be excited (greater than 6.2 ev) that excitation is unlikely except in

extremely powerful systems. In Op the three most easily excited electronic

states are the a A (O.98 ev) , the "b S

"^ (l-65 ev), and the A^S "^ (^.^T ev).

Of these, the first two have extremely small cross sections for direct excitation hy electron collisions [Schulz and McDowell, 1962]-^, so that we need to

3 +

discviss only the excitation of the Ay state which radiates in the Herzberg

hands. These bands are seen in the night airglow and are rendered difficult

to observe because the energy is distributed into many lines in the band.

However, fairly small amounts of excitation shoiold be detectable with

sophisticated techniques

This leaves the D (I.96 ev) and S (^.IT ev) states of atomic oxygen.

The former would be undetectable because it is strongly deactivated by collisions with 0_ at altitudes below about l^i-O km. These calculations do not

apply in any practical situation for altitudes above l40 km because the dominant energy loss term there would be that due to diffusion out of the exciting

beam, an effect which is not included in the calculation of the distribution

function. 0( S) is the strongest line of the nonaal night airglow and

* These states are, of course, excited to some extent; but with the method of

excitation considered here, we find it will never, at any power less than that

which is required for breakdown, be possible to excite a detectable amoimt of

radiation from them.

76
variations of intensity should "be readily observed if excitation is accomplished. This leaves then, as the two most readily observed emissions (A'^r ) and 0( S) Taoth of which require more than k ev for excitation. The generation

of electrons of this energy is particularly difficult in air "because there is

a very large cross section for energy loss to the vihrational states of Np

for all electrons between ahout l.T and h- ev [Schulz, 1959]- It is of interest

to know, however, just what power would be required in order to obtain detectable intensities. In figures la and lb, we plot specific rates of excitation

for the above two states, as well as two body attachment rates andoionization

rates, as a function of the two previously defined reduced powers. In order

to obtain the excitation rate at a particular power and height, one needs to

mtiltiply the value given in the figure by electron density and constituent

density. The cross section for the excitation of the Op(A S ) is estimated

from the data of Schulz [1962], that for the excitation of 0( S) from the

theoretical calculations of Seaton [1953] As an example let us suppose that

we wish to know the power required to excite 100 rayleighs* of 55TT a radiation

(from 0( S) in a slab of the ionosphere between 90 and 100 km) . Then this

means that we must excite 100 atoms per cubic centimeter per second, which will

later radiate. If we assume that there are on the average 500 electrons/cm

and 2 X 10 oxygen atoms/cm in the slab, then we would expect to see approx1 -13

imately 100 rayleighs for"^ specific excitation rate of 0( S) equal to 10

-21

which requires from figure la a reduced hf power of 3 -3 x 10 watts seconds.

This results (using ^a.) in a required power flux of 8 x 10 watts/cm , if

6 2 * The rayleigh is defined as 10 emissions per cm colimm through the

radiating medium.

77
we are considering a transmitter at 50 megacycles. Using (6a) it is found

that 9 X 10 watts are required for an isotropic radiator. If we have an

antenna gain of 10 , we then need 9 megawatts transmitted from the antenna.

The very large power required for minimum detection conditions indicates that

experiments resxalting in the excitation of optical radiation may not he economically feasihle unless the installation can he justified for other applications.

An additional consideration should he made at this point. The calculations made here yield average excitation rates while the power is tiomed

on, and after a steady state has heen estahlished. The time required to estahlish equilihrivim is approximately equal to the product of the average fractional energy loss per collision (averaged over all types of collisions) and the

collision frequency. For air the average energy loss is not constant with

_3

energy, hut for rough calcxolations may he assumed to he of the order of 10 .

For the convenience of the reader and those who migiht wish to extend the calculations to include ahsorption we plot in figures 2a and 2b the average collision frequency, normalized to one particle/cm and the average energy

against the two previously defined reduced powers. The relaxation time will

vary from the order of a microsecond at 60 Ian height to the order of a millisecond at 100 km height. Transmitters must, therefore, he operated with pulse

lengths long compared to these times for steady state to he achieved. For the

higher heights this puts severe requirements on the transmitter hecause of the

long p\ilses required.

One other parameter which must te considered for the case of 0( S) is

that the state has a mean life of the order of one second. For this reason

the effect of pulsed operation will he to reduce the ohserved chsinge in intensity hy the duty cycle. It is to he noted that this does not mean that the

78
average power flux is the important quantity. The extremely non-linear

nature of the intensity vs power curve indicates that the transmitter should

"be operated at as high a peak power as possible and then for as high a duty

cycle as is compatible with this condition.

5. Modification of the Ionosphere by Attachment.

Another possible experiment of interest is the modification of the atmosphere by attaching the electrons to molecules. The problem has been discussed in some detail by Molmud [196if] for attachment by three body attachment,

Og + e + M - Og" + M. (7)

The rate of this process is probably the easiest one to change with

transmitted rf power because there is a maximum in the rate at about 0.1 ev

average electron energy. [Chanin, Phelps, and Biondi, 1959]*

There is a second process which occurs at higher energies. This is

dissociative attachment of Op, by the following reaction,

Og + e - 0" + 0. (8)

This process has a threshold at k ev with a maximum at 6.3 ev [Schiilz, I962]

We may calc^Ilate this rate in the same manner that one caJ.c\alates attachment

rates. These rates are given in figures la and lb. In the example previously

discussed, if we were exciting 100 rayleighs of the 5577 A line we would expect electrons to attach at a rate of 3^ per second. The electrons would effectively disappear at this rate until they are re-released either by collisional detachment or by the reverse of process (8) . These rates are not well

known and it might be that an ionospheric experiment would be one of the best

with which to determine their rates.

79
6. Breakdown in the Ionosphere

The general technique of calciilation used here can, in principle, be

used to compute the power required to "breakdown" the ionosphere. In the

lower ionosphere one may, to a large extent, ignore diffusion losses and

may therefore compute the rate of attachment as a function of power and the

rate of ionization as a f-unction of power. As the ionization rate "becomes

the larger the ionization density will grow exponentially until some limiting process sets in. The details of an approximate treatment of some of

these processes is given hy Lomhardini [I963]. In addition he gives comparisons to experimental data which indicate the order of the accioracy of the

calculations. Specific rates of ionization as a function of (P ), _ and

(P )^ „ are shown in figures la and Ih.

T. F Region Emissions

In the investigation of possible modification of airglow intensities

another possibility of interest presents itself in terms of the F region

emission from 0(l)) at 63OO A. This radiation is presimiably due to the two

step recombination of by means of

and

NO"^ + e - N + O(^). (10)

The rate of (9) is the slower and controls the emission rate so that the

intensity of 63OO A radiation is proportional to the product of the and

Np densities.

Farley [I962, I963] has shown that by using power at the plasma frequency it is possible with reasonable power to heat (~ 25O kw) the electrons

80
enough that the enhanced diffusion decreases the electron and ion density

hy ahout ten percent or more. This should, in turn, decrease the intensity

of 6300 A radiation hy ten percent and serve as an experimental check on

the recomhination process in the F region. It should he noted that the

mean life of the 0( L)) state is 100 sec, so that a transmitter which will

perform the described experiment must he operated cw.

8. Conclusions

The requirements on large power transmitters which can he expected to

yield ohservahle amoimts of optical radiation have been expressed quantitatively for the two most likely radiations. We find that the size of the

required installations is so large that such experiments may not he economically feasible at the present time.

9

Acknowledgements

The author wishes to recognize the very considerable contributions

of Mrs. J. D. Droppleman in the performance of the calculations and the

preparation of the figures.

81
References

Bailey, V.A. (1959), Some possible effects caused "by strong gyrowaves in

the ionosphere. J. Atmos. Terret. Fhys. 1^, 229-32ii-.

Carleton, N.P., and Lawrence R. Megill (1962), Electron energy distrihution

in slightly ionized air under the influence of electric and magnetic

fields, Hiys. Rev. 126, 2089-2099.

Chanin, L.M., A.V. Phelps, and M. Biondi (1959)^ Measurements of the attachment of slow electrons in oxygen, Phys. Rev. Let. 2, 3^^-

Clavier, P. A. (1961), Man-made heating and ionization in the upper atmosphere,

J. Appl. Phys. 32, 5TO-5TT and erratum 2651.

Farley, D.T. (1963), Artificial heating of electrons in the F region of the

Ionosphere, J. Geophys. Res. 68, ^01-^1-13.

Lomhardini, P.P. (1963), Non-linear Conference, WBS, Boiilder, Colorado

Megill, Lawrence R., and J.H. Cahn, {is6k) , Calculation of electron energy

distrihution f-unctions in the ionosphere, to he published.

Megill, Lawrence R., and N.P. Carleton {196k), Excitation hy local electric

fields in the aurora and airglow, J. Geophys. Res. 69, 101•122.

Molmud, P. (1963), Non -linear Conference, KBS, Boulder, Colorado

Schiilz, G.J. (1959)^ Measurement of excitation of N , CO, and H , hy electron

impact, Phys. Rev. II6, llkl-llk-J.

Schxilz, G.J. (1962), 0^, CO, and CO hy electron ijnpact, Phys. Rev. 128,

1T8-

Schulz, G.J., and J.T. McDowell (1962), Excitation of vibrational and electronic levels in Op hy electronic impact, Phys. Rev. 128 , I78-

Seaton, M.J. (1953)^ The Hartree-Fock equations for continuous states with

application to electron excitation of ground configuration terms of 01,

Phil. Trans. Roy. Soc. 2k^

,

kSS.

82
10-9

(Pr)hf

Figure la. This figure shows the rate of excitation of 0( D) (2)^ thip rate of

attachment to 0„, (3), the rate of excitation of O^Ca-^I; ***), and (k)

the rate of ionization as a fimction of reduced power with high

frequency excitation. To obtain an absolute rate the ordinate should

be multiplied by the constituent density and the electron density.

The power flux in watti.'. cm" may be had by multiplying the abscissa

by (2jtf) where f is the frequency of excitation. These cvorves should be used if u < u) ± uu, .

83
10-10

10"

10-12 -

10-13 -

2)

10-14 -

10-15 -

IO-'6 -

10-17

1

10-34 10-33

(Pr)

10-32

If

Figure lb. This figure shovs the same quantities as figure la except that they

are plotted vs the reduced power for low frequency excitation. In

this case the abscissa should be multiplied by the square of the

total number density in particles per cm^ to yield watts cm^. These

curves shoiild be used when U > cju ± ou and for the gyro- resonance

condition. '-'

^h
AVERAGE ENERGY (ev)

u —

Figure 2a. The abscissa is as defined in fig. la while the ordinates are the

average collision frequency (curve 2) jier neutral particle in sec'l and the average

energy in ev (curve l). The averages are over the calculated distribution functions.

85
AVERAGE ENERGY (ev)

"D _

>^c X 10-8

Figure 2b. Die absrlsaa Is as defined In fig. lb while the oitllnates are as defined In fig. 2a.

86
Alteration of the Electron Density of the Lower

Ionosphere with Ground Based Transmitters

Pietro P. Lombardini

Moore School of Electrical Engineering

University of Pennsylvania

Using computational techniques as described by Carleton and

Megillj and Megill to calculate the difference between rate of attachment of electrons to 02'and the ionization rate in air as a function of

electric field, the power required to create breakdown in the ionosphere

has been calculated. These calculations are compared with extrapolated miicrowave breakdown data and found to agree satisfactorily.

The propagation of the primary beam is altered by the electrons

created by it. This gives origin to a non-linear problem, which is

very difficult to deal with in the general case. However, a steady state

approximate solution has been found, considering diffusion in one

dimension. The solution is applied to the design of a possible atmospheric experiment.

87
1. INTRODUCTION

The possibility o£ altering the characteristics of the lower

ionosphere utilizing ground-based transmitters has been considered

by several investigators. In particular, studies have been made concerning the achievement of a decrease in electron density [1], an

increase in the electron collision frequency [ 2] , and the excitation of

air glow [ 3., 4] . Little attention, however, has been given to utilizing

ground-based transmitters to achieve an increase of electron density.

The aim of this paper is to investigate the paramters involved in obtaining such an increase.

The first step of the present work involves the definition of breakdown" as created by a high frequency electromagnetic wave in a boundless,

slightly ionized homogeneous medium. Experimental data for these

conditions of breakdown are not available,' however, the electric field

necessary to initiate breakdown (breakdown threshold) may be obtained

in either of the following two ways. One way consists in performing

direct laboratory measurements of breakdown in finite enclosures, and

then scaling the results to the open air case. The other way consists

of computing the electron distribution function in the mixture of gases

considered (N^, O^ and O) and in the presence of the electric field.

This distribution function computed by utilizing the experimental data

for the various cross sections involved is in turn used to evalute the

"electron attachment and ionization frequencies" (i. e. , v and v. respectively). The breakdown threshold, E, (defined as that field for

which V. = V )^Tnay therefore be calculated.

The breakdown threshold is shown to decrease for certain altitudes

at a faster rate than that specified by the inverse of the distance law

which applies to VHF waves in the far field of the transmitter. From

this, it is concluded that breakdown can be achieved if the proper power

is available. Furthermore, it is shown that once breakdown is obtained,

88
the steady state electron density is critically dependent upon the

difference of the altitude rate of changes of the breakdown field

and the inverse of the distance field.

Because of this dependence on the altitude rate of change of

the exciting field, greater electron densities may be achieved using

a sufficiently large antenna such that breakdown occurs in the near

field.

A characterization of the attachment -ionization process is shown

to indicate an altitude for which the optimum breakdown efficiency

exists.

2. CHARACTERIZATION OF THE BREAKDOWN THRESHOLD

ELECTRIC FIELD

A VHF radiowave traveling upward in the D or in the lower E

region is assumed to be idealized as a plane wave propagating in a weakly ionized, quasi-homogeneous plasma. The breakdown conditions

applicable may therefore be identified with those pertaining to an

electric field of constant orientation and amplitude in a weakly ionized

air. In general, the breakdown condition of an electric discharge is

defined by the electric field (breakdown threshold, Er ) for which

equilibrium is achieved between the gain of electrons due to ionizing

collisions, and the loss due to the various mechanisms which remove

electrons from the discharge region (e. g. , attachment, recombination,

and diffusion). Because of the absence of walls, and the weak ionization

for the case considered, the breakdown is predominantly controlled by

attachment, while diffusion has a negligible role. Conditions of this

kind are rarely encountered in the laboratory where the walls confining

the discharge play a considerable role in removing electrons. Brown

and Rose [ 5] have shown that the experimental data of microwave breakdown threshold in air can be characterized by a curve of ^" versus

P

the product pL, where E rr is the "effective" breakdown electric field

defined by the formula

89
^V + CO

where.

V = average electron collision frequency

(0 = radian frequency of wave

p

= air pressure

L = breakdown gap spacing .

This curve flattens out for increasing values of pL approaching the

3 asymptotic value of 3 x 10 V/m x (mm of Hg). Introducing this

value into Equation (1), considering a collision frequency as a function

of pressure given by 5. 3 x lo" p (cps) [ 6] , one obtains the following

simple equation relating the r m s value of the breakdown threshold

field in air with the pressure,

E^ = 3 X 10^

°

A] 5. 3 X 10^ p

Specification of breakdown conditions in open space may also be

obtained utilizing a different approach. That is, the energy distribution

function of the electrons under the influence of an oscillating electric

field in a mixture of N2, O^ and O may be calculated. In this approach,

the experimental data introduced are the collision cross sections of the

electrons with the above mentioned components of air. Megill and

Carleton [ 7] have given a detailed analysis of the problem of determining

the electron energy distribution function under these conditions. Megill

has constructed a computer program for the IBM 704 of Raytheon, which

gives the electron energy distribution function, f (u), for the interval of

electron energy, u, from 0.01 to 20. e. v. , and having neutral particle,

density, temperatures, geomagnetic field, frequency, and the electric

90
field intensity as parameters. The solution applies to a situation

in which the total number of electrons does not change. Providing

that the rate of change of the total nximber of electrons is small,

and incorporating perturbation techniques, the solution f (u) may be

used for evaluating the rate of increase of electrons due to ionizing

collisions.

1

S N.

J

u f (u) a.

J (u) du

' JO

- r°°l/2

V. = -i r-^ (3)

{-^) I u f (u) du

or the rate of decrease of electrons due to attachment,

CO

£ N

a

. Juf(u) o-.^(u) du

o

V = -i 2 (4)

,ml/2 r°° 1/2,. . , (-5-) ^ f (^) ^^

where N.. a.^ and <r. . are the concentration, ionization cross section

i' il lA

and attachment cross section for the i^^ component of air considered

(i. e. , Np, O^, and O) and m is the electron mass. In the actual computations, the only attachment phenomenon considered is the dissociative

attachment, O, + e ^ O + O". Ionization and attachment collision

frequency obtained in this way may be characterized as a function of the

electric field intensity E, holding all other parameters constant, v^

grows rapidly for moderate fields and quickly tends to saturate, while

91
V. starts a quick rise at higher field intensities. The crossing of

the two curves (v. = v ) corresponds to the breakdown threshold Et^ . Using the atmospheric model presented in the Handbook for Astronautical

Engineering [8], extensive computations have been made for the following choice of paraireters: fr: 50 mc/9, geonnagnetic field = 5 x 10" weber/m

and concentrations and temperatures corresponding to the height interval,

55 - 105 KM. The corresponding values of E^^ as a function of altitude

are shown in Figure 1 by the solid curve. In the same figure, the

dotted curve shows the Ei^ altitude variation as obtained using Equation (2).

The good agreement between the two methods of approach is apparent

from this figure. The collision frequencies given by (3) and (4) are

required for evaluating the rate of change of the total number of electrons

given by

TT = (^i - V N (5)

Computed values of v. - v are plotted in Figure 2 as a function of the

1 3. excess of electric field, A E, and the same range of paramters previously

A E specified for Figure 1. The relationship between v. - v and ^=— is

1 a h,^

approximately linear.

In Figure 3, v. - v is plotted vs. height for constant electric field

above threshold breakdown. This figure shows that there exists an

altitude at which the creation of new electrons occurs with a maximum

efficiency. This optimum height corresponds to the condition for which

the average collision frequency approximately equals the frequency of

the electric field; i. e. ,

1

V (h, E, ) ^ f . (6)

92
850 -I

800-

750 -

700-

650-

600-

550-

2 300-

>

4 50-

400 -

360-

300 -

230 -

200-

I 50-

FIG. 1 BREAKDOWN THRESHOLD ELECTRIC FIELD

VS. ALTITUDE FOR A FREQUENCY OF 50 mc/s

100

COMPUTER RESULT

EXTRAPOLATION OF EXPERIMENTAL RESULTS

55 60 65 -r

70 n I I 75 80 85 ALTITUDE (km)

90 95 100 105

93
10 5_

10^

10*

10 I .

10'

60km 67.5 kM

ftl.Sk*

f = 50 ine/s

.5kiii

THE DIFFERENCE OF THE IONIZATION AND

ATTACHMENT COLLISION FREQUENCIES AS A

FUNCTION OF THE PERCENT INCREASE

OF EXCESS ELECTRIC FIELD ABOVE BREAKDOWN

0.1 1.0

E^

9J*
FIG. 3 THE DIFFERENCE OF THE IONIZATION AND

ATTACHMENT COLLISION FREQUENCIES AS

A FUNCTION OF ALTITUDE FOR THE FIELD

STRENGTHS INDICATED

10 -

290 V/M

200 V/M

f = 50 mc/s

lO'-

n—I—I—I—I—I—I—I—I—I—I—I—I—I—

70 80 ALTITUDE (km)

95
Figure 4 shows the relationship between frequency and altitude as obtained by Equation (6), using for v, McDonald's [6] approximation.

To each of these frequencies corresponds a breakdown threshold

electric field, which may be determined using Equation (2). The

ground-based upward radiating transmitter needed for obtaining the

above defined electric field, E, , at the altitude given by Figure 4,

must have a power -gain product, PG, which for free wave propagation

is given by

PG = 4 IT h

2 ^b

(•7)

where r| is the wave impedance in space.

In Table 1 are tabulated breakdown-transmitter parameters. In

the last two columns of the table gain and power are given, assuming

that the 1, 000 foot Arecibo paraboloid antenna is used as the transmitter.

tABLE I- TRANSMITTER PARAMETERS TO ACHIEVE BREAKDOWN

FREQUENCY FOR THEORETICAL POWER NEEDED

BREAKDOWN OPTIMUM IONIZATION POWER X GAIN GAIN USING USING ARECIBO

ALTITUDE EFFICIENCY ARECIBO ANTENNA

(km) (mc/s) (WATT) ANTENNA (MEGA WATT)

60 1,000 1.2 lO" 10-^ 1 20

65 500 3.8 10'* 2.5 10® 150

70 250 I0"» 6.0 1 0» 165

75 100 2.0 10" 10' 200

80 40 3.5 I0'2 1.6 10* 220

8 5 15 5,0 10" 2.2 lO' 230

96
FIG. 4 FREQUENCY OF MAXIMUM IONIZATION EFFICIENCY

VS. ALTITUDE

1,000 -I

50 0-

200

100 -

50 -

20

—T

60

I 70 "T80 90 ALTITUDE ( km)

97
3. MAXIMUM ATTAINABLE INCREASE OF ELECTRON DENSITY

In the previous section, the conditions required for obtaining

breakdown have been examined. The problem is now that of determining the maximum electron density which may be achieved using

radio waves. For this purpose, consider a VHF wave propagating

vertically upward. Under normal ionospheric conditions, a wave

in this band of frequencies is negligibly absorbed by the ionosphere.

However, when the electric field strength of the wave becomes larger

than the breakdown threshold, the electron density will increase and

hence the wave may suffer an exponential attenuation. In free space,

the electric field of the wave may be expressed as:

E (x) = Eq $ (x) (8)

where E„ is the amplitude of field intensity at an altitude, h , from

the ground, where the source is assumed to be located. Here, x is

the local vertical coordinate in a system whose origin is at a height,

h , and $ (x) represents the functional variation in the absence of

absorption. In the "far field" of an antenna.

$(x) =

X +

(9)

IT

o Along the axis of a spherical antenna excited in phase and focused at

h , $ (x) is given by

Sin TT D

8 \ h

$(x) =

TT D

(10)

i8 X h

where D is the antenna diameter and X the wavelength.

98
If the propagation takes place in ionized air, the wave is

subjected to exponential absorption. When the electron densityis sufficiently low, the differential power absorbed along the wave

path is

d

HIT

E^ (X)

2

2 r°°

= - N(x)

(eE(x)) 3/2 , , 8f(u) , i i—i^— I u a(u) -jr-i—' du m J

* ' 9u

/°°u'/2f(u) du

(11)

where N (x) is the electron density at the altitude h + x and a (u) is

given by.

a (u) = -^ 2" V + w

(12)

disregarding the earth's magnetic field. Proceeding to integrate Equation

(11) in the interval x = to the arbitrary distance x.

E (x) = Eq $ (x) exp [•/ A (E, x) N (x) dx (13)

where the quantity A (E, x) is a slowly varying function of E and x.

99
In order to simplify the problem, let us for the moment disregard diffusion. Assume that for the radio waves propagating upward with some spatial variation, ^

(x) fe. g. , inverse of the distance

law] breakdown conditions are reached at the altitude h, . The physical

situation may be explained from Figure 5 where the spatial variation

of the electric field of the wave and the breakdown threshold variation are plotted as a function of altitude. These two curves cross at two

altitudes, h, and h^- Consider a point located at the altitude h, + x, where the spatial variation field curve lies above the breakdown curve. Here the intensity of the wave will change with time because of the

attenuation due to the time increasing electron density. Hence, assume

the spatial variation curve of Figure 5 represents the established situation

at time, t = 0. It follows at this point (h, + x) that the electron density

will start increasing exponentially with time according to the solution

of Equation (5),

N (X, t) = Nq exp |^(v. - v^)

t"J (14)

where N^ is the previously existing electron density. It has already

been pointed out that the quantity (v. - v ) is critically dependent upon

1 3. the quantity A E = E - E^ (see Figure 2). Substituting Equation (14) into

Equation (13),

AE = Eq $ (x) exp - A (x) Nq exp [(v. - v ) t] dx E,(x) .(15)

Equation (15) does not take into account any mechanism limiting the

growth of electron density as a function of time, such as losses due

to diffusion or recombination. Therefore, for increasingly large

100
FIG. 5

SKETCH OF THE VARIATION OF THE FREE SPACE

FIELD STRENGTH (DASHED LINE), AND OF THE

BREAKDOWN THRESHOLD (SOLID LINE) VS. ALTITUDE

ALTITUDE101
times, satisfaction of Equation (15) requires that AE approaches

zero. In the limit condition,

Lim

t —? 00

/

A (E, x) N (x, t) dx = - In

o Eq 45 (x) (16)

N = Lim N (x, t) = . ;„ . max ^

'

' A (E, x)

t —^00

(X) aE^(">

x

f(3Er Eb(x) (17)

An important observation of Equation (17) is that the steady state

distribution of N is critically dependent upon the slopes of the field

and the breakdown threshold curves.

4. THE DIFFUSION PROBLEM

Equation (17) was obtained ignoring diffusion. The rigorous solution

taking diffusion into account is unknown. However, for steady state, if

the electric field and the density of the neutral components of air are

slowly varying functions of the coordinates, the electron balance may

be approximated by the equation:

D (E. X) ^ ^M^ = - N (x) ( V. - V J (18)

dx

where.

102
oo

J

u^/^ v"-^ (u) f (u) du

o D(E, x)= (19) /oo

u f (u) du

is the diffusion coefficient which is assumed to be a slowly varying

function of the altitude and the electric field.

Equations (13) and (18) describe completely the steady state

situation for N (x) and E (x). The system can be solved using the

boundary conditions suggested by physical considerations (see Figure 6).

The boundary conditions will thus be chosen at two altitudes h, and h^,

one below and the other above the crossing of the breakdown threshold.

d N For this altitude, the value of N and -3— are assumed to be known. d X

Examining Equations( 13) and (18) we note that while A (E, x) and

D (E, x) are slowly varying functions of E and x, the function v. - v may undergo large ^anges for small changes of either variable (as

shown in Figure b/. Thus, Equation (18) is very sensitive to small

changes of E. This renders the system of Equations (13) and (18) highly

nonlinear and difficult to deal with. This system has been solved

numerically by Albertoni, Bocchieri, and Daneri [9l with the help

of an IBM 7090 computer.

5. NUMERICAL EXAMPLES

Two cases are considered here which employ the results of the

previous section. The first case deals with the "far field" of the transmitter. The parameters selected are,

f = 50 mc/s

^1 = 71 KM

Eo, = 196 V/m .

103
12,000-

10,000-

8,000-

6,000-

4,000-

2,000-

to.

2,000-

FIG. 6

VARIATION OF THE DIFFERENCE OF THE

IONIZATION AND ATTACHMENT COLLISION

FREQUENCIES AS A FUNCTION OF THE

FIELD STRENGTH

8, 000-1

f = 50 MC/S

V/M

-I

300

lOit
In Figure 7 is plotted the electron density distribution as obtained

using Equation (14) (curve on the right). Also plotted (curves on

the left; note the change in scale) are the solutions of the system of

Equations (13) and (18). The two curves, which correspond to the

different "trial" parameters chosen in the numerical integration,

represent the physical solution as long as they coincide. Beyond

the point of bifurcation they fail to indicate a physical solution. The

region of coincidence could be prolonged with a better choice of parameters. However, the data in Figure 8 show that beyond the region

considered, the approximation obtained neglecting diffusion Equation (14)

is adequate.

The second case refers to a hypothetical transmitting antenna

large enough such that the breakdown field is located at its focus.

Here it is assumed that the antenna is a spherical basin whose surface

is radiating in phase. The law of variation of the field intensity near

the focus is thus assumed to be given by Equation (10), where

IL^_^ = 1.7x 10" Vm.

8X h^

The other parameters are chosen such that in absence of electronic

absorption, the electric field radiated reaches a maximum of 200 v/m

at the altitude of 78 KM, where the focus of the antenna is assumed

to be located. In Figure (8) is plotted the electron density variation

with altitude obtained using Equation (17). Also plotted are the

solution of the system of Equations (13) and (18) which is valid as a physical solution up to the point of bifurcation.

105
^ N -N

y^ _ <o /

K / _IO

1

ri^ /

o /

Q /

d^^ // u. / - ^

< /

10 h. /

«s / q: 2 / - o \ /

u. > / N z / N o ^ / — o> / •- " / m

" ' ^m ^. N

Qi - \ /

S^¥ \/ ^ Q « h 1/

>-^t K

O

b e 2 /

Z in Z

_o

UJ u <

=> - u

_ o

c

111 o \ -s - a _ \ ~ I b hi ^^ _o

Id {/) ^v,. o

o J < ^^^-^ o_ LJ O ^^---^^^^^ °- N

r^ ^~~~..^

H _o ^"

e) ^^^-^^^^^^

iZ —.......^^ _ o '1 111 III 1 - at • N

o o o c1

"TM (£-"') N

1

io6
FIG. 8 ELECTRON DENSITY DISTRIBUTION FOR CASE 2

[f = 50 mc/s, E( h, ) =200 V/M, E^ = 156 V/M, NEAR

FIELD OF transmitter]

^12-1

2 10* 40 X (METER) — ho= 78,000 m

107
6. CONCLUSIONS

It has been shown that a radiowave radiated by a ground based

transmitter may achieve breakdown in the D or lower E ionospheric

regions. For each altitude, a frequency exists at which there corresponds an optimum ionization efficiency. The maximum electron

density which can be obtained critically depends upon the altitude

rate of change of both the breakdown field and the free space transmitted field strength. This characteristic automatically limits the

maximum obtainable electron density which may be achieved in the

far field of an antenna. This limitation may, however, be overcome

by having breakdown take place in the near field.

108
ACKNOWLEDGEMENT

The study described in this paper has been totallysupported by the Raytheon Company, for which the author is a Consultant. Acknowledgement should also be given to all

the people who have worked on this project: L. R. Megill

(National Bureau of Standards, Boulder, Colorado),

S. Albertoni, P. Bocchieri, A. Daneri (Applications of

Scientific Research, Milan, Italy), C. F. Eaton, and

R. C Gunther (Raytheon Company).

109
REFERENCES

[ 1] Molmud, P. , "Use of Radio Transmitters to Decrease

D-Region Electron Density, " 1963. (in this issue)

[ 2] Farley, D. T. Jr. , "Artificial Heating of the Electrons

in the F -Region of the Ionosphere, " J. Geoph. Research,

68, 1963, pp. 401.

[ 3] Bayley, V. A. , "Some Possible Effects Cuased by Strong

Gyrowaves in the Ionosphere, " J. Atmos. Terr est. Phys.

14, 1959, pp. 229.

[4] Megill, L. R., "The Excitation of Optical Radiation by High

Power Density Radio Beams," (in this issue)

[5] Brown, S. C., Rose, D. , "Microwave Gas Discharge

Breakdown in Air, Nitrogen and Oxygen, " J. Appl. Phys.

28, 1957, pp. 561.

[ 6] Mac Donald, Gaskell, and , Gitterman, "Microwave

Breakdown in Air, Oxygen, and Nitrogen, " Phys. Rev.

130 , 1963, pp. 1841.

[7] Carleton, N. P., Megill, L. R. , "Electron Energy

Distribution in Slightly Ionized Air Under the Influence of Electric

and Magnetic Fields, " Phys. Rev» 126, 1962, pp. 2089-

[8] Handbook of Astronautical Engineering, McGraw-Hill,

1961, pp. 2-9.

[ 9] Albertoni, S. , Bocchieri, P. , Daneri, A. , Conference

on Applications of Electronic Computers to Space Problems,

Florence, 12-13 September 1963.

«l

110
Nonlinear Effects in Radiation Generation Through

the Coupling of Electron Beams with Diffraction Gratings

Winfield W. Salisbury, Sc.D.

Chief Scientist

Varo, Inc., 2201 Walnut St., Garland, Texas

The experimental studies of the Smith-Purcell effect

which we have been pursuing in the Research Laboratory of Varo,

Inc. have led to some unexpected results. The Smith-Purcell

effect is the radiation of visual light by free electrons

passing rapidly across the surface of a metallic diffraction

grating. The early experiments reported by Smith and Purcell

used electrons up to a little over 300 kv accelerating potential or having beta values in the region between 0.75 and

0.8. Our experiments have involved somewhat lower energies

with beta values from about 0.4 to 0.75 or, roughly, in the

range from 50 to 250 kv of accelerating potential. The wavelength formula given by Smith and Purcell is

A = D (-^ - Cos 0) . Where ^ is the observed wavelength of radiation, D is the

grating spacing; /^is the ratio of the electron velocity to

the velocity of light; and 9 is the angle getween the direction

of the electrons and the line of observation.

Ill
This formula has been found to be correct to a high degree of accuracy with one exception. The gratings which are

normally available from optical work produce a spectrum rather

than a single wavelength in a given direction. This spectr\im

can be accurately predicted if the profile shape of the grating

lines is known. The alteration in the wavelength formula is

simple. Dividing a grating spacing D by a number N, which may

take on integral values, will give the correct spectrum, providing the integers represented by N are chosen to represent

the index numbers of the Fourier series terms which represent

the grating profile. Thus, a grating profile which is symmetrical around a perpendicular to the average surface of the

grating will require only odd terms in its Fourier series, and

only the odd integers will occur as appropriate values of N.

In the case of a blazed grating, however, both even and odd

terms occur in the representative Fourier series. Harmonics

out to the seventh have been observed in this way.

These facts of wavelength and the occurance of a null in

the radiation intensity in the direction where

p = Cos e

are compatible with a simple electrostatic image theory in

which the electron and its supposed positive image below the

grating surface form a moving oscillating dipole, which accounts

for the radiation. (Figure l). Several attempts to make an

intensity theory based on this radiation have not so far proved

very successful.

An experiment regarding the effect on electron distance

from the grating sheds some light on this lack of agreement,

and indicates the possibility of known linearities and a more

complex requirement for theoretical explanation. Electron

beams .025 mm thick have been used for most of our experiments.

112
This thickness ii; approximately 500 wavelengths for green

light, which is frequently observed. For this reason, it has

been expected that only a small fraction of the electrons

very close to the grating would be effective in producing radiation. A series of experiments to test this, however, have

given strong indications to the contrary. For example: An

adjustable electron-slit gate was devised and the light intensity at a fixed angle measured as a function of the gate

position. Surprisingly, the intensity proved to be an almost

linear function of gate position in such a way as to indicate

that electrons at the surface of the beam furthest from the

grating were almost as effective in producing light as those

next to the grating. (Figure Z .) Movements of the grating away from the beam, however,

extinguish the light with movements of an entirely different

order of magnitude. This seems to indicate that the beam of

electrons acts as a total entity, rather than a mere summation of the action of individual electrons. This should

be explainable by some form of non-linearity. The usual ionosphere-type equations, however, do not seem adequate, as the

plasma frequencies for the electron beam lie in the ordinary

radio frequency region for the highest electron densities so

far used (up to about 10 electrons per cubic centimeter)

The possibility has been considered that a plasma is

formed by the electron beam in the residual gas in the vacuum.

—ft —ft However, operation m vacuums from 10 torr to 10~ torr have

produced no appreciable change in intensity. Operation with

pure argon gas between 10 and 10 torr pressure produce an

increase in intensity of about 20% at the high pressure region.

It seems reasonable to attribute this effect to increased

current density in the beam, because of the effect of space

charge neutralization, which is commonly known as gas focussing.

113
A word or two about experimental equipment seems desirable

for the benefit of those who may wish to follow this research

more in detail. In order to permit the study of electron

beams up to 300 kv of accelerated potential, an experimental

apparatus has been built in which the potential for acceleration

is split in half by a concentrically enclosed division which

is only open through the electron slit and some small alignment jig holes. (Figure 3. ) This is in the form of concentric

stems supported vertically by a tower of Pyrex glass insulators

in oil. The electron emitter which is in the innermost surface

mounted on a supporting stem consists of a ridge on an indirectly heated nickel cylinder. This ridge is placed between

beam forming electrodes which are at the same potential as the

emitter. The beam forming electrodes are sloped at a 67° angle

to the electron beam in the manner of a Pierce gun designed

for the formation of a sheet electron beam. A control slot

is mounted ahead of this and is connected to an insulated

power supply through the stem so that its potential with

respect to the emitter can be independantly varied from

up to 5,000 volts with respect to the emitter. This is

surrounded by a separate support stem and an outer sphere which

is separately insulated. The whole arrangement is inserted

into a ground potential chamber which contains the grating

mounting and adjustment micrometers. The electron deflector

and catcher system and an adjustable ground potential slot,

which is the final focussing electrode for the electron beam

formation, are shown clearly in the photograph. The innermost stem containing the emitter is operated at the full

negative potential through which the electrons are accelerated. The support stem and spherical shield is operated

at half this potential by means of a resistance voltage divider.

llif
The electron beam focus is sufficiently good so that the electrons collected by the slot in this outer shield cause a

negligible shift in the dividing potential. The usual arrangement is with the emitter blocked off by emitter potential

shields so that a beam, 1 cm wide, is formed. This sheet beam

is 1/4 mm thick. Good focussing is observed for total accelerating potentials of 50 kv and upwards. Various harmonics of

the grating profile are observed in the form of visual light

with accelerating voltages from 20 kv upwards. Visual light is

expected to be observed with even lower potentials when a new

grating, 76,000 lines per inch, is placed in the test position.

Gratings of various profiles having 15,000, 20,000, 35,000,

45,000, and 55,000 lines per inch have been used with results

which conform very closely to those expected insofar as the

harmonic content and directional pattern of light intensity

is concerned. Observed intensities have been considerably

greater than should be expected on the basis of interaction

with a layer of electrons having a depth of one wavelength

or less for the emitted radiation, or having a depth of the

order of one grating spacing.

A grating profile as determined from electron microscope

photographs, showing the dimensions in microns, is shown in

Figure 4. The Fourier series which represents this profile

is as follows:

f (X) = 0.2263 - 0.0656 cos

^

^ - 0.0674 sin ^

^

+ 0.0184 cos ^ ^^ - 0.0645 sin

^^^ + 0.0264 cos -^ ^^

+ 0.0204 sin ^ ^^ + 0.00137 cos ^ ^

^

+ 0.02025 sin ^ ^'^

+ 0.01284 cos ^ T^ + 0.004028 sin ^ if ^

115
A picture from the light from this grating is shown in Figure 5,

This picture was obtained by photographing the electron beam

through an 8,000 lines per inch transmission grating. The extra

images on each side each represent a harmonic present in the

emitted light. Beginning at the outside, away from the center

image and counting inward toward it, we have the 4th, 5th, 6th,

and faintly, the 7th harmonic of the Fourier analysis of this

grating as separate images. This photograph is quite striking

shown in the original color. The color of the light also depends on the angle of observation as indicated by the original

wavelength equation. This is due to the Doppler effect introduced by the fact that the apparent source of radiation moves

with the velocity of the electrons. This is not very troublesome, however, as the intensity at low angles in the forward

direction may be hundreds of times that seen at an angle of

90 to the grating for example, and thousands of times the

light seen at the minimum angle where some radiation may appear due to the observing lens taking in Azimuth angles other

than . Experiments are continuing and any suggestions regarding

possible theoretical explanations will be gratefully received.

This work is sponsored by Varo, Inc., Garland, Texas, and

by the United States Air Force, Aeronautical Systems Division,

Wright Patterson Air Force Base, Dayton, Ohio.

References

Smith, Stephen J. , and Edward M. Purcell (Nov. 15, 1953), Visible

Light from Localized Surface Charges Moving Across a Crating,

Letter, The Physical Review 9^, No. 4, 1069.

Salisbury, Winfield W.^ United States Patent, Number 2, 634, 372,

issued in April 1953.

116
Panofsky, WolfgaJig K. H, , and Melba Phillips, Classical Electricity

and Magnetism, Para. 19-4, 306-307. (Addison-Wesley Publishing

Company, Cambridge, Mass., 1955).

Palocz, Istvan, A Leaky Wave Approach to Cerenkov and Smith-Purcell

Radiation, Dissertation submitted for the degree of Doctor of

Philosophy at the Polytechnic Institute of Brooklyn. (Ann Arbor,

Univ. Microfilms, 1962),

117
©00 ©0© ©

1, ELECTRON PATH -A^A/W^-

GRATING SURFACE

Figure 1.

118
0.5 • INTENSITY

ARBITRARY UNITS

GATE POSITION

(MILLS)

Figure 2.

119
Figure 3 This photograph shows the electron beam focal electrodes in the background

-and next forward the diffraction grating in its adjustable mountings, then the

small permanent magnets which deflect the electrons downward out of the line

of sight, and in the immediate foreground the watercooled electron catcher which

is made of silver and tungsten to absorb the avoidable X-rays due to collected

high energy electrons. This is viewed through a 14" optical glass port.

120
H

uaaooo

in

)30 • HU

tao OU

n)

II o

II r ou

121
Figure 5 Four spectral images on each side of the central image. The colors in tne original

photograph of the spectral images are, starting from the outside in sequence inward,

red, green, blue, and violet. They represent the 4th, 5th, 6th, and 7th harmonic

of the Fourier Series representing the grating profile.

122 GPO 84 1 - 146