By including the relativistic mass change in the motion of electrons gyrating in a slightly inhomogeneous field, it is possible to account for the cyclotron echoes observed by Hill and Kaplan

Gould, R. W.

Kegel, W. H.

Caltech Authors - Main


ON THE THEORY OF
PULSE STIMULATED RADIATION FROM A PLASMA
W. H. Kegel and R. W. Gould
Technical Report No. 30 November 1965
Research sponsored by the
U. S. Office of Naval Research
Contract Nonr 220(50)
CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California
Volume 19, number 7 PHYSICS L E T T E R S  15 December 1965 
ON T H E  T H E O R Y  O F  P U L S E  S T I M U L A T E D  R A D I A T I O N  F R O M  A P L A S M A  # 
W. H. KEGEL ## and R. W. GOULD 
California Institute of Technology, Pasadena, California 
Received 12 November 1965 
The a im of this l e t te r  i s  to show that the r e l a t iv i s t i c  m a s s  effect in a cold p l a sma  in an inhomogeneous 
magnet ic  f ield gives r i s e  to an echolike phenomenon [1] with al l  the c h a r a c t e r i s t i c s  of the exper imen ta l ly  
observed  echoes r epor t ed  by Hill and Kaplan [2]. 
We cons ider  a p l a sma  in a s l ight ly inhomogeneous magnet ic  f ield exci ted by two shor t  pu lses  at the 
e l ec t ron  gyrofrequency,  the t ime separa t ion  of the pu lses  being r .  We ask for the r ad ia t ion  af ter  the 
second pulse.  We t rea t  the problem in the s ingle  pa r t i c le  approach. 
The rad ia ted  in tens i ty  depends on the phase co r r e l a t i ons  between the different  pa r t i c l es .  More p re -  
c isely ,  the t ime va r i a t ion  of the rad ia t ion  is  given by 
N -2 N a/]} 2 
• (t)=(l~=lrl)= l~=lrlexp{i[c~lt+ , (1) 
where N is the number of particles considered, r l is the Larmor radius of the lth particle, w / is its 
gyrofrequency, and ~l its phase at time t = 0. @ (t) is normalized such that it is unity when all particles 
have the same phase. From (1) one sees that it is sufficient to determine phase differences. For this it 
is convenient to consider the phases in a coordinate system rotating with the average gyrofrequency Wc" 
We neglect the initial energy of the electrons (cold plasma approximation). At the end of the first 
pulse all particles have in this approximation the same energy and the same phase. We now account for 
the field inhomogeneities by attributing a different gyrofrequency to each electron. (This implies that 
the inhomogeneities are perpendicular to the field lines.) We assume a distribution g(7/) over the differ- 
ent gyrofrequencies, where y = AinhW c is the deviation from the average gyrofrequency due to the in- 
homogeneities. The relative phase of a particle at a time ~ after the first pulse is 
¢(n t ,  0 = ~l ~"  (2) 
After the second pulse the pa r t i c l e s  have different ene rg ie s  according  to their  phase at the onset  of this 
pulse .  This give r i s e  to addit ional  d i f ferences  in  the gyrof requenc ies  due to the r e l a t i v i s t i c  m a s s  effect. 
With v 2 << c 2 we have af ter  the second pulse 
~*wl c = A ( B -  r~2/R 2) + "l (3) 
with A = ~3R2/2c2, R being the L a r m o r  r ad ius  af ter  the f i r s t  pulse and r~  that af ter  the second pulse.  
B is  an a r b i t r a r y  constant  which defines the par t i c le  with r e spe c t  to which a * ~ c  is  measured .  The r e l a -  
tive phase of a pa r t i c le  at Ume t, now m e a s u r e d  f rom the second pulse,  is  then given by 
vt ( t )  = q , t ( t=  o) + A *.,Zct (4) 
(p~(t=0) and r~  of a pa r t i c le  depend on i ts phase at the onset of the second pulse and are g iven by 
"Y*12/R 2 = 1 + D 2 + 2D cos [ ¢ ( , / ,  ~) - q~o] (5) 
¢ ~ ( t  = 0) = ~ o  + & (6) 
This work was sponsored by the U.S.Navy, Office of Naval Research. 
$$ On leave from the Institute fffr Plasmaphysik, Garching bei Mffnchen, Germany. 
531 
Volume 19, number 7 PHYSICS L E T T E R S  15 December 1965 
sin/31 = (R/r~) s in  [(P(~T/, T) - (go] , ('7) 
where D gives the s t reng th  of the second pulse re la t ive  to the f i r s t  and q~o the phase of the e lec t r i c  field 
of the second pulse r e l a t ive  to a pa r t i c le  with (p(T) = 0. With these re la t ions  we have, after  the second 
pulse:  
@(t) ~ I fd ,7  g(~?){O exp[if(ff)] + exp[ifl(7/)]}l 2 (8) 
f(TI) = ~o + A t [B  - 1 - D 2 - 2D cos (~?T- q~O)] + T/t (9a) 
fl(~?) = ~?T - ~0 o + f(T/) . (9b) 
As there is  a cosine in the exponent ial  functions,  we make an expansion into Besse l  ~unctions" 
exp[if(~/)] = ~ Jk(2ADt) exp{i[~/(t+kT) - (k-1)~o o - ½/~r + ~1}. (10) 
k ~ -  oo 
stands for al l  t e r m s  not depending e i ther  on 7/or the index of the Besse l  functions.  F rom (10) it fol- 
lows that there a re  max ima  of $(t) at t imes  
trn = m T  , m = 1, 2, 3 , . . .  (11) 
(P(tm) ~ (1 + 0 2 ) - l { I D J m ( 2 A D t  m 12 + Igm_l(2AOtm) 12}. (12) 
If the a rgument  of the Besse l  funct ions is large we have $(tm) ~ 1/t  m. When col l i s ions  a re  taken into 
account (12) is  to be mul t ip l ied  by an exponent ial  function. The shape of the rad ia t ion  peaks is  the square 
of the Four i e r  t r a n s f o r m  of g(77). 
As an example of their  expe r imen ta l  r e s u l t s  Hill and Kaplan [2] give an osci l lograrn  showing two radi- 
at ion max ima  at t imes  ~- and 2~-, the in tens i ty  ra t io  being 8 : 1. This ra t io  is  e s sen t i a l ly  de te rmined  by 
col l i s ions ,  as 1- is  of the order  of the r epor ted  decay constant .  
In the case of three  exci t ing pu lses  Hill and Kaplan find an echo at a t ime ~- after  the thi rd  pulse. The 
dependence of i ts  in tens i ty  on T (time between the second and the thi rd  pulse) tu rns  out to be de te rmined  
by the ine las t ic  co l l i s ions  only. 
If one t rea t s  the t h r ee -pu l se  case at f i r s t  in the co l l i s ion less  approximat ion analogous to the two- 
pulse case, one obtains  an express ion  for $(t) now consis t ing  of sums  over products  of Besse l  functions 
mul t ip l ied  by an exponent ial  function of the kind 
exp{i[~/(t + (k + rn +n) T + (1 + m) T + #~m]}, (13) 
where k, l, m, n a re  summat ion  indices .  F r o m  this one concludes that rad ia t ion  maxima a r i s e  at t imes  
tKL =K + L T ,  =~K,L = 0 , 1 , 2 , . . .  (14) 
t now being m e a s u r e d  f rom the th i rd  pulse.  If one now a s s u m e s  that the phases  of the e lec t rons  are ,  due 
to e las t ic  col l i s ions ,  r andomized  between the second and thi rd  pulse,  but the individual  energ ies  are  
p rese rved ,  one mus t  cons ider  the phase 7/T in (13) as a s ta t i s t i ca l  quanti ty over which one mus t  inte-  
grate.  This in tegra t ion  leads to the cancel la t ion  of all  t e r m s  with L ¢ 0, while those with L = 0 survive .  
The e s sen t i a l  difference with the two-pulse  case is  that, after  the second pulse ,  there is informat ion  
s tored  not only in the phases ,  but a lso in the energy dis t r ibut ion,  which is  not des t royed by e las t ic  col- 
l i s ions .  
If the p l a sma  d imens ions  are  of the order  of the wavelength (or la rger ) ,  the k -vec to r s  of the excit ing 
pulses  have to be perpend icu la r  to the magnet ic  field. Otherwise,  pa r t i c les  excited at different  phases  
can exchange the i r  p laces  and spoil  by that the genera ted  phase cor re la t ions .  If the in i t ia l  t empera tu re  
of the p l a sma  is too high, the rad ia t ion  max ima  vanish  [3]. 
References  
1. R.W.Gould, Physics Letters 19 (1965} 477. 
2. R.M. Hill and D. E. Kaplan, Phys. Rev. Letters 14 (1965) 1062. 
3. W.H.Kegel, to be published. 
532 
UNCLASSIFIED
California Institute of Technology 
Pasadena, California
ON THE THEORY OF PULSE STIMULATED RADIATION FROM A PLASMA
Technical Report, December 1965
KEGEL, Wilhelm H. 
GOULD, Roy W.
December 15, 1965 2 3
Nonr 220(50)
Technical Report No. 30
Qualified requestors may obtain copies of this report from DDC.
By including the relativistic mass change in the motion 
of electrons gyrating in a slightly inhomogeneous field, it is 
possible to account for the cyclotron echoes observed by Hill and 
Kaplan.
UNCLASSIFIED
UNCLASSIFIED
CYCLOTRON RADIATION 
PLASMA RADIATION
ECHOES
UNCLASSIFIED


English

On the Theory of Pulse Stimulated Radiation from Plasma


ON THE THEORY OF
PULSE STIMULATED RADIATION FROM A PLASMA
W. H. Kegel and R. W. Gould
Technical Report No. 30 November 1965
Research sponsored by the
U. S. Office of Naval Research
Contract Nonr 220(50)
CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California
Volume 19, number 7 PHYSICS LETTERS 15 December 1965 
ON THE THEORY OF  PULSE ST IMULATED RADIAT ION FROM A PLASMA # 
W. H. KEGEL ## and R. W. GOULD 
California Institute of Technology, Pasadena, California 
Received 12 November 1965 
The aim of this letter is to show that the relat iv ist ic  mass effect in a cold p lasma in an inhomogeneous 
magnetic field gives r i se  to an echolike phenomenon [1] with all the character is t ics  of the experimental ly 
observed echoes reported by Hill and Kaplan [2]. 
We consider a p lasma in a sl ightly inhomogeneous magnetic field excited by two short pulses at the 
electron gyrofrequency, the t ime separat ion of the pulses being r .  We ask for the radiat ion after the 
second pulse. We treat the problem in the single part ic le approach. 
The radiated intensity depends on the phase corre lat ions between the different part ic les.  More pre-  
cisely, the t ime var iat ion of the radiat ion is given by 
N -2 N a/]} 2 
• (t)=(l~=lrl)= l~=lrlexp{i[c~lt+ , (1) 
where N is the number of particles considered, r l is the Larmor radius of the lth particle, w / is its 
gyrofrequency, and ~l its phase at time t = 0. @ (t) is normalized such that it is unity when all particles 
have the same phase. From (1) one sees that it is sufficient to determine phase differences. For this it 
is convenient to consider the phases in a coordinate system rotating with the average gyrofrequency Wc" 
We neglect the initial energy of the electrons (cold plasma pproximation). At the end of the first 
pulse all particles have in this approximation the same energy and the same phase. We now account for 
the field inhomogeneities by attributing a different gyrofrequency toeach electron. (This implies that 
the inhomogeneities are perpendicular to the field lines.) We assume a distribution g(7/) over the differ- 
ent gyrofrequencies, where y = AinhW c is the deviation from the average gyrofrequency due to the in- 
homogeneities. The relative phase of a particle at a time ~ after the first pulse is 
¢(nt ,  0 = ~l ~" (2) 
After the second pulse the part ic les have different energies according to their phase at the onset of this 
pulse. This give r i se  to additional differences in the gyrofrequencies due to the re lat iv ist ic  mass effect. 
With v 2 << c 2 we have after the second pulse 
~*wl c = A(B-  r~2/R 2) + "l (3) 
with A = ~3R2/2c2, R being the Larmor  radius after the f i rst  pulse and r~ that after the second pulse. 
B is an arb i t ra ry  constant which defines the part ic le with respect  o which a*~c is measured.  The re la-  
tive phase of a part ic le at Ume t, now measured from the second pulse, is then given by 
vt(t)  = q,t(t= o) + A *.,Zct (4) 
(p~(t=0) and r~ of a part ic le  depend on its phase at the onset of the second pulse and are given by 
"Y*12/R 2 = 1 + D 2 + 2D cos [¢( , / ,  ~) - q~o] (5) 
¢~(t  = 0) = ~o + & (6) 
This work was sponsored by the U.S.Navy, Office of Naval Research. 
$$ On leave from the Institute fffr Plasmaphysik, Garching bei Mffnchen, Germany. 
531 
Volume 19, number 7 PHYSICS LETTERS 15 December 1965 
sin/31 = (R/r~) sin [(P(~T/, T) - (go] , ('7) 
where D gives the strength of the second pulse relat ive to the f i rst  and q~o the phase of the electr ic field 
of the second pulse relat ive to a part ic le with (p(T) = 0. With these relat ions we have, after the second 
pulse: 
@(t) ~ I fd,7 g(~?){O exp[if(ff)] + exp[ifl(7/)]}l 2 (8) 
f(TI) = ~o + At[B - 1 - D 2 - 2D cos (~?T- q~O)] + T/t (9a) 
fl(~?) = ~?T - ~0 o + f(T/) . (9b) 
As there is a cosine in the exponential functions, we make an expansion into Bessel  ~unctions" 
exp[if(~/)] = ~ Jk(2ADt) exp{i[~/(t+kT) - (k-1)~o - ½/~r + ~1}. (10) 
k~-  oo 
stands for all te rms not depending either on 7/or the index of the Bessel  functions. From (10) it fol- 
lows that there are maxima of $(t) at t imes 
trn =mT , m = 1, 2, 3 , . . .  (11) 
(P(tm) ~ (1 + 02) - l{ ID Jm(2ADt  m 12 + Igm_l(2AOtm) 12}. (12) 
If the argument of the Bessel  functions is large we have $(tm) ~ 1/t m. When col l is ions are taken into 
account (12) is to be mult ipl ied by an exponential function. The shape of the radiat ion peaks is the square 
of the Four ier  t ransform of g(77). 
As an example of their exper imental  resu l ts  Hill and Kaplan [2] give an osci l lograrn showing two radi- 
ation maxima at t imes ~- and 2~-, the intensity ratio being 8 : 1. This ratio is essential ly determined by 
col l is ions, as 1- is of the order of the reported decay constant. 
In the case of three exciting pulses Hill and Kaplan find an echo at a time ~- after the third pulse. The 
dependence of its intensity on T (time between the second and the third pulse) turns out to be determined 
by the inelastic col l is ions only. 
If one treats the three-pulse case at f i rst  in the col l is ionless approximation analogous to the two- 
pulse case, one obtains an expression for $(t) now consist ing of sums over products of Bessel functions 
multipl ied by an exponential function of the kind 
exp{i[~/(t + (k + rn +n) T + (1 + m) T + #~m]}, (13) 
where k, l, m, n are summation indices. From this one concludes that radiat ion maxima ar ise at t imes 
tKL =K +LT ,  =~K,L =0,1 ,2 , . . .  (14) 
t now being measured from the third pulse. If one now assumes that the phases of the electrons are, due 
to elastic col l is ions, randomized between the second and third pulse, but the individual energies are 
preserved,  one must consider the phase 7/T in (13) as a stat ist ical  quantity over which one must inte- 
grate. This integration leads to the cancel lat ion of all terms with L ¢ 0, while those with L = 0 survive. 
The essent ia l  difference with the two-pulse case is that, after the second pulse, there is information 
stored not only in the phases, but also in the energy distr ibution, which is not destroyed by elastic col- 
l is ions. 
If the plasma dimensions are of the order of the wavelength (or larger),  the k-vectors of the exciting 
pulses have to be perpendicular to the magnetic field. Otherwise, part ic les excited at different phases 
can exchange their places and spoil by that the generated phase correlat ions.  If the init ial  temperature 
of the plasma is too high, the radiat ion maxima vanish [3]. 
References 
1. R.W.Gould, Physics Letters 19 (1965} 477. 
2. R.M. Hill and D. E. Kaplan, Phys. Rev. Letters 14 (1965) 1062. 
3. W.H.Kegel, to be published. 
532 
UNCLASSIFIED
California Institute of Technology 
Pasadena, California
ON THE THEORY OF PULSE STIMULATED RADIATION FROM A PLASMA
Technical Report, December 1965
KEGEL, Wilhelm H. 
GOULD, Roy W.
December 15, 1965 2 3
Nonr 220(50)
Technical Report No. 30
Qualified requestors may obtain copies of this report from DDC.
By including the relativistic mass change in the motion 
of electrons gyrating in a slightly inhomogeneous field, it is 
possible to account for the cyclotron echoes observed by Hill and 
Kaplan.
UNCLASSIFIED
UNCLASSIFIED
CYCLOTRON RADIATION 
PLASMA RADIATION
ECHOES
UNCLASSIFIED


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On the Theory of Pulse Stimulated Radiation from Plasma

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