J. Plasma Physics (2000), vol. 63, part 4, pp. 329–341. Printed in the United Kingdom

# 2000 Cambridge University Press

329

Self-induced erosion and spectral breaking ofhigh-power microwave pulses

D. ANDERSON," A. KIM,# M. LISAK" and K. MADSEN$

"Department of Electromagnetics, Chalmers University of Technology,S-412 96 Go$ teborg, Sweden

# Institute for Applied Physics, Nizhny Novgorod, 603600 Russia$National Defence Research Establishment, Box 1165, S-581 11 Linko$ ping

(Received 2 March 1999 and in revised form 12 October 1999)

Abstract. An analysis is made of the phenomenon of self-induced erosion andspectral breaking of ionizing high-power microwave pulses propagating in a gas.The analysis describes in an analytically explicit and physically clear way theconsistent interaction between the microwave pulse and the self-inducedbreakdown plasma. In particular, it clarifies, both qualitatively and quan-titatively, the mechanisms behind the pulse erosion and the spectral breakingphenomenon, i.e. the splitting of the pulse spectrum into a redshifted and ablueshifted peak as observed in numerical simulation results as well as inexperiments.

1. Introduction

High-power microwave pulses propagating in air tend to suffer pulsedistortion, so-called pulse erosion, due to absorption andor reflection in theself-induced breakdown plasma (Kuo and Zhang 1991; Kim et al. 1995;Koretzky et al. 1998). Different aspects and applications of this phenomenonhave been studied in a number of different investigations. An interestingfeature that has been given much consideration recently (Koretzky et al. 1998)is the fact that the rapid rise of the electron density in the self-inducedbreakdown plasma tends to upshift the pulse frequency. In fact, it has evenbeen suggested (Gildenburg et al. 1989) that the pulse may propagate withoutcutoff reflection, since the pulse frequency may conceivably increase morerapidly than the plasma frequency, thus preserving the underdense plasmasituation. This self-induced spectral shift of high-power microwave pulses hasrecently been investigated in a series of beautiful experiments (Koretzky et al.1998), which have also been compared and found to agree with numericalsimulation results.

However, although good agreement is obtained between the experimentalresults and the computer simulations, we feel that a good, physically clear,description is lacking. The purpose of the present work is to analyse theinteraction between the microwave and the self-induced breakdown plasma interms of a simple but self-consistent physical model that clarifies the mainphysical processes involved in the pulse erosion phenomenon including thespectral dynamics. In particular, we shall give an explicit analytical solutionthat clearly describes the main features of the pulse erosion phenomenon: an

330 D. Anderson et al.

initial part unaffected by erosion, a subsequent part involving a rapid decay ofthe pulse amplitude due to attenuation in the breakdown plasma, and finally atrailing part with a plateau or ledge formation where the self-induced ionizationin the reduced microwave field is balanced by losses due to attachment.Depending on the relative magnitude of the physical parameters, the differentparts may be more or less dominating; for example, the trailing part mayexhibit only the monotonically decaying behaviour if the pulse erosion is notstrong enough to suppress the pulse amplitude to the saturation valuecharacteristic of the plateau. The corresponding solutions for the phase of themicrowave clearly display the spectral characteristics, including the spectralbreaking observed in the experiments.

However, the interpretation of the spectral dynamics is found to be differentfrom previous descriptions. It is found that the initial part of the pulse (wherethe pulse erosion is small) suffers little spectral shift, but the frequency increasesrapidly during the ‘rampdown’ phase of the amplitude, and may either reacha higher but constant frequency or return to its original value, depending on thedegree of pulse erosion. Thus, in the case of monotonically increasing frequency,the total spectrum can be viewed essentially as the spectrum of a pulseconsisting of two parts with different amplitudes and different frequencies. Thissimple picture reproduces the main features of the experimental results as wellas of the simulations. In particular, it has been observed that the spectrummay split into two parts : the ‘spectral breaking’ phenomenon where onepart is redshifted whereas the other is blueshifted. These features are directconsequences of the above ‘two-pulse ’ picture, including the redshifting, whichis found to be simply the consequence of the interference between the two parts.The same features may also arise in the case of the nonmonotonically increasingfrequency variation, where the frequency first increases and then returns to itsinitial value.

2. Dynamical equations

A number of more or less sophisticated and detailed models have beendeveloped for describing the interaction between the microwave pulse and theself-generated breakdown plasma. The relevant equations are essentiallythe wave equation for the microwave field and the continuity equation for theelectron density of the breakdown plasma. The propagation properties ofthe microwave are determined by the plasma electron density, and con-versely the electron density is determined by an interplay between themicrowave-induced collisional ionization and various loss mechanisms such asdiffusion and attachment, which tend to deplete the electron density. Thus aconsistent treatment requires the simultaneous solution of the wave equationand the equation of continuity.

In the subsequent analysis, we shall assume that wave reflection in the self-generated plasma can be neglected and that the pulse erosion is due toabsorption in the breakdown plasma. This assumption is legitimate in twolimits (cf. Koretzky et al. 1998): (i) when the gas pressure is low and the powerof the microwave is not too high above the breakdown threshold, and (ii) whenthe pressure is high (of the order of atmospheric pressure). In these cases, the

Erosion and spectral breaking of microwave pulses 331

wave equation can be reduced to a first-order equation describing only theforward-going wave. It is convenient to transform into a coordinate systemmoving with the group velocity of the wave and to use the reduced time τ,which is the time measured relative to the arrival time of the pulse, i.e. τ¯t®xv

g, where t is the ordinary time, x is the distance of propagation, and v

gis

the group velocity of the wave. In this system, the wave equation is reduced to(cf. Gildenburg et al. 1989; Kuo and Zhang 1991; Kim et al. 1995; Koretzky etal. 1998)

2ik!

¦E

¦x­k#

!

n

nc

01­iν

ω1E¯ 0, (1)

where E is the slowly varying envelope of the microwave, k!

and ω are thewavenumber and frequency respectively of the microwave, n and n

care

the electron density and the critical electron density respectively (at the criticalelectron density, the corresponding plasma frequency equals the microwavefrequency), and ν is the collision frequency between the electrons and theneutral gas molecules.

In (1), the electron density n depends on the strength of the microwaveelectric field in a way determined by the continuity equation, namely

¦n

¦τ¯ [ν

i(rEr)®ν

l]n, (2)

where νi(rEr) is the ionization frequency due to electron collisions with neutral

particles and νldenotes the loss frequency due to attachment andor diffusion.

The ionization frequency depends strongly on the electric field, and can bemodelled with different degrees of sophistication, but a very good (and oftenused) approximation for not too large electric field strengths is (cf. MacDonald1966)

νi(rEr)E ν

i! 0 rErE

th

1#β

, (3)

where νi!

is the ionization frequency corresponding to the electric field strengthE

th. The loss frequency depends much more weakly on the electric field, and will

be assumed to be independent of the electric field strength (cf. MacDonald1966).

Writing the electric field E in terms of real amplitude A(x, τ) and phase#(x, τ) according to E¯A exp(i#), inserting this into the wave equation (1), andseparating the real and imaginary parts, we obtain two coupled equations:

¦A

¦x¯®

k!

2

n

nc

ν

ωA, (4a)

¦#

¦x¯

k!

2

n

nc

. (4b)

From (2)–(4), we realize that A and n form a closed system of equations and,having solved this, the phase variation is directly determined by (4b). Thus the

332 D. Anderson et al.

system determining the consistent interaction between the microwave pulseand the concomitant breakdown plasma is given by

¦A

¦x¯®

k!

2

n

nc

ν

ωA, (5a)

¦n

¦τ¯ [ν

i(A)®ν

l]n. (5b)

3. Solution of dynamical equations

It is convenient to normalize and rewrite (5a, b) as follows. We introduce

B¯ 0 A

Eth

1#β

,

where Eth

is defined as corresponding to the threshold field for breakdown, i.e.

νi(E

th)¯ ν

l,

n

nc

¯N,

xa ¯βk

!xν

ωU x, τ- ¯ τν

i!U τ.

The system can then be written in the simple form

¦B

¦x¯®NB, (6a)

¦N

¦τ¯ (B®1)N. (6b)

From the solution of (6a, b), the phase is then determined by the relation

¦#

¦x¯

N

2β. (7)

In order to solve (6a, b), we introduce the ‘sweep-up’ integral ρ¯ ρ(x, τ) definedby

ρ¯&x

!

N(x«, τ) dx«, or equivalently¦ρ

¦x¯N. (8)

Equation (6a) can then be integrated immediately to give

B(x, τ)¯B!e−ρ(x,τ), (9)

where B!is the initial value of the function B(x, τ), i.e. B

!¯B(0, τ)¯ rA(0, τ)r#β.

Using (8) and (9) in (6b), we obtain the following equation (after integratingonce)

¦ρ

¦τ¯B

!(1®e−ρ)®ρ3F(ρ), (10)

subject to the initial condition ρ(x, 0)¯ ρ!(x)' 1. It does not seem possible to

find an exact solution of (10) in general. However, an exact solution can be

Erosion and spectral breaking of microwave pulses 333

obtained in the limit when the losses are negligible, and a simple approximatesolution can also be obtained in the general case. We shall solve the equationin these two cases separately.

3.1. Negligible losses

For short pulses, breakdown will only occur when the microwave power issignificantly above the continuous-wave breakdown threshold. In this situation,we have ν

i(rEr)( ν

l, and (10) simplifies to

¦ρ

¦τ¯B

!(1®e−ρ), (11)

which is trivially solved to yield

ρ(x, τ)¯ ln(1­exp9&τ

!

A#β(0, τ«) dτ«­c

!(x):* , (12)

where c!(x) is determined by the initial condition ρ(x, 0)¯ ρ

!(x), i.e.

c!(x)¯ ln(eρ

!(x)®1). (13)

The solution is then given explicitly as

A(x, τ)¯A(0, τ) exp9®ρ(x, τ)

2β : , (14a)

N(x, τ)¯¦ρ

¦x, (14b)

#(x, τ)¯ρ(x, τ)

2β, (14c)

where ρ(x, τ) is given by (12).The solution simplifies further if we assume that ρ

!(x)' 1, i.e. small initial

electron density and negligible initial damping of the wave. Then

c!(x)¯ ln[ρ

!(x)] (15)

and

ρ(x, τ)¯ ln(1­ρ!(x) exp9&

τ

!

A#β(0, τ«) dτ«:* . (16)

Without serious restriction, we can assume a rectangular pulse shape, i.e.

A(0, τ)¯1

23

4

A!

for 0% τ% τp,

0 otherwise,(17)

and

n!(x)¯

1

23

4

n!

for x& 0,

0 otherwise,(18)

which implies thatρ!(x)¯n

!x for x& 0, (19a)

&τ

!

A#β(0, τ«) dτ«¯A#

β

!τ for 0% τ% τ

p. (19b)

334 D. Anderson et al.

The solution can now be written in the explicit form

ρ(x, τ)¯ ln(1­n!xeA#

β

!τ) (20a)

A(x, τ)¯A!(1­n

!xeA#

β

!τ)−"/#β, (20b)

#(x, τ)¯1

2βln(1­n

!xeA#

β

!τ). (20c)

3.2. Losses important

When the pulse amplitude decays owing to erosion, the ionization frequencydecreases rapidly and the loss frequency will ultimately become important(provided the pulse is long enough). In this case, the full form of thecharacteristic function F(ρ) introduced in (10) must be retained and it does notseem possible to find an exact solution of (10). Nevertheless, an explicitapproximate solution can be obtained for this case as follows.

The qualitative form of F(ρ) is easily inferred from (10) to be essentiallyparabolic between the two zeros of F(ρ), which are ρ¯ 0 and ρ¯ ρkrespectively, where

ρk¯

1

23

4

2(B!®1)

B!

if ρk' 15B!®1' 1,

B!

if ρk( 15B!®1( 1.

(21)

We conclude that a simple useful approximation retaining the importantfeatures of the function F(ρ) can be given as

F(ρ)E (B!®1) ρ 01®

ρ

ρk1 , (22a)

ρk¯ ρk(τ)EB#

!®1

B!

. (22b)

With this approximation for F(ρ), and assuming B!(τ) to be independent of τ,

i.e. a rectangular pulse, an explicit solution for ρ(x, τ) is easily obtained as

ρ(x, τ)¯ρk(τ)

2 91­tanh0B!(τ)®1

2[τ®τ

!(x)]1: , (23)

where τ!(x) is determined by the initial condition ρ(x, 0)¯ ρ

!(x), i.e.

τ!(x)¯

2

B!®1

artanh01®2ρ!

ρk1 . (24)

Thus the solution in this case can be summarized as follows:

ρ(x, τ)¯ρk(τ)

2 91­tanh0B!(τ)®1

2[τ®τ

!(x)]1: , (25a)

A(x, τ)¯A(0, τ) e−ρ(x,τ)/#β, (25b)

#(x, τ)¯ω

ν

ρ(x, τ)

2β, (25c)

Erosion and spectral breaking of microwave pulses 335

where ρk(x) and τ!(x) are given by (22) and (24) respectively.

We note that similar results for the solution of the coupled system (5) havebeen given in Semenov (1982) and Anderson et al. (1986) in connection withstudies of a self-consistent study of an ionization wave produced by microwavebreakdown in atmospheric air.

3.3. Qualitative discussion of pulse erosion dynamics

It is instructive to first discuss qualitatively the properties of the obtainedsolutions in order to gain a better insight into the physics involved in the pulseerosion and spectral breaking phenomena.

From (20) we infer that significant pulse erosion occurs when n!x exp(A#

β

!τ)

" 1, i.e. for times τ such that

τ" τk¯®ln(n

!x)

A#β

!

. (26)

Consequently, a dynamical breakdown criterion for the microwave pulse can beformulated simply as τ

p¯ τk. When τ

p! τk, the trailing part of the pulse will

be more or less suppressed, and in the region of significant pulse erosion theamplitude decays exponentially as (assuming unsaturated decay, i.e. negligibleelectron losses)

A(x, τ)EA

!

(n!x)"/#β

e−A#β

!τ/#

β. (27)

An important consequence of the solution given by (20) is that the pulsefrequency becomes chirped by the self-induced ionization. The chirp frequencyΩ is determined by

Ω¯¦#

¦τ¯

1

2β

n!xA#

β

!eA#

β

!τ

1­n!xeA#

β

!τ, (28)

and we note that in the region of significant pulse erosion, the pulse frequencybecomes upshifted to the constant frequency Ω

maxgiven by

Ωmax

¯A#

β

!

2β. (29)

In order to understand the basic qualitative features of the pulse dynamics, wecan divide the pulse into three parts (i) 0% τ% τk, (ii) τk% τ% τ

!, and (iii) τ

!% τ% τ

p, where the pulse characteristics can be approximated as

(i) 0% τ% τkA(x, τ)EA(0, τ), (30a)

ΩEn!x

2β' 1; (30b)

(ii) τk% τ% τ!

A(x, τ)EA

!

(n!x)"/#β

e−A#β

!τ/#

β, (31a)

ΩEΩmax

¯A#

β

!

2β" 1; (31b)

(iii) τ!% τ% τ

p A(x, τ)EA!e−ρk/#β, (32a)

ΩE 0. (32b)

336 D. Anderson et al.

The qualitative features of the pulse erosion process can easily be inferred from(30)–(32). For slightly overcritical situations and not too long propagationdistances, τk and τ

!are both very long (τ

!(x)( τ

p). Furthermore, ρE ρ

!' 1,

and the amplitude as well as the phase of the microwave are essentiallyunaffected. When τkE τ

p, pulse erosion begins to affect the very trailing part

of the pulse, which clearly indicates that the dynamic breakdown condition canbe taken as τkE τ

p.

When τk! τp! τ

!, pulse erosion will become an important effect in

determining the transmitted pulse form. The damping factor ρ(x, τ) will remainvery small for times τ such that τ! τk, but will then increase rapidly to makethe amplitude decay exponentially. However, when τ

!! τ

p, the damping factor

will saturate at the larger value ρk around τE τ!, after which it will stay

approximately constant for τ!! τ! τ

p.

The corresponding qualitative form of the pulse amplitude can then beinferred to be as follows. The pulse should consist of a first part unaffected bypulse erosion, a middle part involving a rapid decrease of the amplitude, andfinally a suppressed plateau region, features that are clearly seen in most of theexperiments (cf. e.g. Koretzky et al. 1998). The physical explanation of theplateau region is now obvious: the absorption of the microwave in the self-generated breakdown plasma decreases the wave amplitude until it becomesequal to the threshold amplitude, where the ionization frequency is equal to theloss frequency.

The corresponding qualitative variation of the frequency should exhibit acharacteristic feature in the form of a locally blueshifted frequency during therapid-decay phase of the amplitude. The consequences of this frequency shiftfor the resulting frequency spectrum of the pulse will be illustrated in moredetail in the next section, but it is illustrative to consider the spectrum from anidealized pulse corresponding to the qualitative picture given above, i.e.

A(x, τ)¯1

23

4

A"

for 0% τ% τ",

A#

for τ"% τ% τ

"­τ

#,

(33a)

Ω(x, τ)¯1

23

4

0 for 0% τ% τ",

Ω"

for τ"% τ% τ

"­τ

#.

(33b)

Obviously, if Ω"is sufficiently large (Ω

"" 1τ

"­1τ

#), the spectrum will consist

of simply the two separated sin# xx# spectra for each pulse. However, if thespectra overlap, interference between the two contributions will give rise to aspectrum that typically contains two peaks: one blueshifted and one redshifted.This mechanism is the qualitative explanation for the observed experimentalspectra. A more detailed and rigorous analysis of the solutions for the pulseerosion and the concomitant form of the spectrum will be given in the nextsection.

4. Quantitative description of pulse erosion and pulse spectrum

Having discussed in a qualitative way the properties of the pulse erosiondynamics, we now turn to a more quantitative analysis based on the explicitsolutions given earlier. The theoretical predictions will be compared with resultsobtained in Koretzky et al. (1998). The high-power microwave pulsepropagation experiments of Koretzky et al. (1998) were performed in a cubic

Erosion and spectral breaking of microwave pulses 337

1.0

0.8

0.6

0.4

0.2

035 40 45 50 55

(a)

1.0

0.8

0.6

0.4

0.2

0–3 –2 –1 2 3

(b)

0 1

Retarded time

Normalized frequency

Spe

ctra

l am

plit

ude

(arb

itra

ry u

nits

)N

orm

aliz

ed a

mpl

itud

e

Figure 1. Pulse form (a) and pulse spectrum (b) when losses are unimportant,i.e. according to (20) and for A¯ 1±1 (other parameters are given in the text).

vacuum chamber filled with dry air. A microwave pulse from a magnetron wasfed into the chamber through a horn at one side of the cube, and thetransmitted pulse was received by another horn at the opposite side, where itwas analysed. The power of the incident microwave pulse was held constant,and the pressure of the chamber gas was varied in the pressure range0±1–1±0 Torr. At these low pressures, the dominant loss mechanism should beelectron diffusion out of the breakdown region. However, we emphasize that,from the point of view of the theoretical considerations, the actual physics ofthe loss mechanism is not important. A spectrum analyser and an oscilloscope

338 D. Anderson et al.

1.0

0.8

0.6

0.4

0.2

035 40 45 50 55

(a)

1.0

0.8

0.6

0.4

0.2

0–3 –2 –1 2 3

(b)

0 1

Retarded time

Normalized frequency

Spe

ctra

l am

plit

ude

(arb

itra

ry u

nits

)N

orm

aliz

ed a

mpl

itud

e

Figure 2. As Fig. 1, but for A¯ 1±41.

were used to record the pulse spectrum and the envelope of the incident as wellas the transmitted pulse.

In Figs 1 and 2, we show two examples of the pulse erosion and thecorresponding spectrum for increasing input power. The curves correspond tothe case when losses can be neglected, and the parameters are chosen tocorrespond to the experimental situation and the simulations presented inKoretzky et al. (1998). In particular, ν

"¯ 0, p¯ 0±1 Torr, τ

p¯ 1 µs, ν¯

3±5¬10*p(Torr) s−", ω¯ 2±1¬10"! rad s−", N!N

c¯ 10−&, and 2βE 3±6. The

figures clearly show the characteristic features observed in Koretzky et al.(1998) for increasing power (A¯ 1±1 in Fig. 1 and A¯ 1±41 in Fig. 2) : theincreasing erosion of the trailing part of the pulse (Figs 1a, 2a) and the

Erosion and spectral breaking of microwave pulses 339

1.0

0.8

0.6

0.4

0.2

015 16 17 21 22

(a)

1.0

0.8

0.6

0.4

0.2

0–5 5

(b)

0

1.2

18 19 20 23 24 25Retarded time

Normalized frequency

Spe

ctra

l am

plit

ude

(arb

itra

ry u

nits

)N

orm

aliz

ed a

mpl

itud

e

Figure 3. Pulse form (a) and pulse spectrum (b) when losses are important, i.e. accordingto (25) and for A¯ 1±3 (other parameters are given in the text).

concomitant development of the spectrum from a small blueshift of the singlespectrum into a double-peaked spectrum, where the peaks are blue- as well asredshifted respectively (Figs 1b, 2b).

In Figs 3 and 4, we show a corresponding example of the situation when lossesbecome important. The physical parameters are the same as in Figs 1 and 2except that p¯ 0±9 Torr and 2βE 5±34 (because of the higher electric fields).For intensities just above breakdown, the behaviour is the same as in Figs 1 and2. However, for increasing intensities and increasing pulse erosion, losses tendto saturate the amplitude decay and start to form a plateau in the trailing part

340 D. Anderson et al.

1.0

0.5

015 16 17 21 22

(a)

1.0

0.8

0.6

0.4

0.2

0–5 5

(b)

0

1.5

18 19 20 23 24 25Retarded time

Normalized frequency

Spe

ctra

l am

plit

ude

(arb

itra

ry u

nits

)N

orm

aliz

ed a

mpl

itud

e

Figure 4. As Fig. 3, but for A¯ 1±55.

of the pulse. The spectra again develop the two-peaked structure characteristicof the experiments and the simulations. The normalized amplitudes in thissequence are A¯ 1±3 (Fig. 3) and 1±55 (Fig. 4).

5. Conclusions

A detailed analytical investigation has been given of the phenomenon of self-induced erosion and spectral breaking of ionizing high-power microwave pulsespropagating in a gas. The analysis describes in an analytically explicit andphysically clear way the consistent interaction between the microwave pulseand the self-induced breakdown plasma, and clarifies, both qualitatively and

Erosion and spectral breaking of microwave pulses 341

quantitatively, the characteristic features observed in numerical simulations aswell as in experiments.

The main features of the pulse erosion phenomenon can be summarized asfollows: the leading part of the pulse is unaffected by erosion, a subsequentmiddle part will exhibit a rapid decay of the pulse amplitude due to attenuationin the breakdown plasma, and finally a trailing part may occur (provided thepulse intensity is high enough), with plateau or ledge formation where the self-induced ionization in the reduced microwave field is balanced by losses due toattachment. Correspondingly, the spectrum of the microwave clearly displaysthe spectral characteristics of a slight blueshift of the spectrum for moderatewave intensities, continuously transforming into the breaking of the spectruminto a redshifted and a blueshifted peak for high intensities.

References

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Gildenburg, V. B., Krupnov, V. A. and Semenov, V. E. 1989 Soviet Tech. Phys. Lett. 14, 738.

Kim, J., Kuo, S. P. and Kossey, P. 1995 J. Plasma Phys. 53, 253.

Koretzky, E., Kuo, S. P. and Kim, J. 1998 J. Plasma Phys. 59, 315.

Kuo, S. P. and Zhang, Y. S. 1991 Phys. Fluids B3, 2906.

MacDonald, A. D. 1966 Microwave Breakdown in Gases. Wiley, New York.

Semenov, V. E. 1982 Soviet J. Plasma Phys. 8, 347.