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NPS 61-87-004 



NAVAL POSTGRADUATE SCHOOL 

Monterey, California 




CERENKOV AND SUB-< 


2ERENK0V 


RADIATION 




FROM A CHARGED 


PARTICLE 


BEAM 




By 






J 


R. NEIGHBOURS 
'and X. K. ! 


, P. R. BUSKIRK, 




4ARUYAMA 






3 March 


1987 






Technical 


Report 





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NAVAL POSTGRADUATE SCHOOL 
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ITLE (Include Security Classification) 
Cerenkov and Sub-Cerenkov Radiation from a Charged Particle Beam 



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3STRACT (Continue on reverse if necessary ana iaentify by block numoer) 

BSTRACT : As a consequence of the relaxation of the phasing condition between the 



oving charge and radiated wave for finite beam path lengths, the Cerenkov peak is 
coadened and the threshold energy is developed which is applicable to charged 
earns consisting if singie point charge or charge bunch of finite size, as well as 
aams consisting of periodically repeated bunches. 



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Cerenkov and Sub-Cerenkov Radiation from a Charged Particle Beam 

John R. Neighbours 

Fred R. Buskirk 

Xavier K. Maruyama+ 

Physics Department 

Naval Postgraduate School 

Monterey, California 93943 

ABSTRACT 
As a consequence of the relaxation of the phasing condition 
between the moving charge and radiated wave for finite beam path 
lengths, the Cerenkov peak is broadened and the threshold energy 
is lowered. A criterion for the threshold energy is developed 
which is applicable to charged beams consisting of single point 
charge or charge bunch of finite size, as well as beams 
consisting of periodically repeated bunches. 



Permanent Aadress - National 3ureau of Stanaaras, 

Zai chersburg, MD 2089° 



INTRODUCTION 

In previous work 1 we have given a form for the Cerenkov 
radiation from periodic electron bunches propagating in a 
homogeneous medium. The method involved construction of the 
Fourier components of the field which in turn led to the Poynting 
vector expressed as harmonics of the basic electron beam 
frequency. The important results were that the Cerenkov cone 
angle is shifted substantially beyond the ordinary Cerenkov angle 
9 C , (cos9 c = (nB) -1 ) and broadened so that a significant fraction 
of the power radiated appears at angles other than 8 C . As either 
frequency or the path length of the beam increases, the cone 
angle was found to approach 9 C with an increasing fraction of the 
total radiation being radiated at that angle. 

In a second paper^, we showed preliminary experimental data 
and the results of calculations for X ana K band microwave 
Cerenkov radiation produced by the electron bunches from an 5 
band ^00 MeV Linac beam propagating in air. These results as 
well as other published-^ and unpublished ones are in suostantial 
agreement with the predictions of reference 1. 

Subsequently we discussed several aspects of the expected 
Cerenkov radiation Ccom an intense electron oeam \ and the 
emission threshold^ for radiation in brief reports. The onset: of 
Cerenkov raaiation was ascribed ^o a relaxation of the phase 
matching condition between the charge and the wave and the effect 
has been investigated theoretically and experimentally in the 
optical region. °'?'8 



-2- 



Recently^ we have corrected an error appearing in the 
appendix of reference 1 and have shown that the radiated energy 
from a single charge bunch has the same form as the radiated 
power from a beam of periodic bunches. Radiation from a periodic 
beam occurs at the fundamental frequency and harmonics, whereas 
radiation from a single charge bunch has a continuous frequency 
distribution. 

The purposes of this paper are: 

(1) to amplify our remarks concerning the effects of a 
finite electron beam path on the sharpness and intensity of the 
radiation pattern. 

(2) to show in detail how the energy threshold for the 

onset of Cerenkov radiation is affected by the electron beam path 
path length. 



-3- 



CHARACTERISTICS OF COHERENT CERENKOV RADIATION 

The total coherent power per unit solid angle, radiated 
at the frequency v by a periodic charged particle beam in 
traveling a finite distance L at constant velocity is^ 

W(v,k) = v 2 qr2 (D 

where v is the fundamental frequency of the beam generator and v 
is a harmonic of v , q is the charge of an individual bunch, Q is 
a constant and R is the radiation function. The bunches in the 
beam are assumed to be rigid, i.e. unchanging in shape and size 
as the beam travels through the medium (usually air) at a 
velocity given by v = 8c . The velocity of light in vacuum is c 
and the velocity of light in the medium is c = c /n where n is 
the inaex of refraction. 

Similarly, 5. the energy radiated per unit solid angle 
within the frequency range dv oy a single bunch of charge q 
traveling a distance L is^ 

E(v,k) dv = QR2dv (2) 

In both cases the constant 2 is 

Q = — £££__ (3) 

Bit 
wnere u is the permeability of the medium, ana che radiation 

function R is given by 

-»- 
R - 2 t n sin 8 I(u) F(k) (4) 

where is the angle between the direction of travel of the 

-4- 



charged particle beam and the direction of propagation of the 
emitted radiation, I(u) is the diffraction function, and F(k) is 
the form factor. In the remainder of this paper we refer only to 
W although the discussion and results also apply to E, since W 
and E have the same spatial distribution. 

It is convenient to measure the finite length of travel of 
the charged particle beam in units of the wavelength in the 
medium of the emitted radiation. Accordingly, the dimensionless 
beam length parameter n which appears in the radiation function 
explicitly, and implicitly as part of the diffraction parameter 
defined below, is defined as the ratio of the path length of the 
charged particle beam to the wavelength in the medium of the 
emitted radiation. 






'he iiff ."action function is 



I(u) 



sin u 



u 



(6) 



where the dimensionless diffraction parameter u depends upon the 



Cerenkov angle given by cos9 c = (nS) 1 , as well as upon the be 
Length parameter ana the radiation emission angle. 



am 



i = irnC(nB) * cos 6 ] 



(7) 



The *ave vector of the emitted radiation is < (k = uj/c), and 

F(k) is the dimensionless form factor. That is, if p(r) is the 
charge distribution of a single bunch, then the Fourier 
components of the charge are 



-5- 



ik -r 



p(k) = J J J p(r) e 1K * r d3 



(8) 



and the form factor is defined by 

p(k) = q F(k) (9) 

For a point charge, F(k) is identically one for all values of k. 

The actual radiator is the medium. Radiation is emitted in 
a cylindrically symmetric pattern about the length of the charged 
beam and with a transverse polarization lying in the plane of the 
beam and the direction of propagation of the radiation. Since R 
depends directly upon n, the strength of the radiation is 
proportional to the square of the path length of the beam. 

When the wavelength of the emitted radiation is long 
compared to the dimensions of an individual charge bunch, F(k) 
is slowly varying with ingle so that the oattern of smitted 
radiation is dominated by the diffraction function, I(u). 
Regardless of the behaviour of F(k), the radiation pattern nas 
zeroes occuring for integral multiples of tt. The largest value 
of (I(u))2 occurs at u = (3=6 C ) with subsidiary maxima at 
u = 1.4303, 2.4590, 3.4709, 4.4774 etc. The maxima of the 
radiation (W or E) are lispiacea from these values by the (sin8)2 
factor, and jy the relatively small variation of (F(k))^. 

Formally, the diffraction function radiation pattern is 
similar to that resulting from diffraction by a single slit in 
which the incoming plans wave is highly oblique, or to an "end 
fire" antenna array. The actual radiation pattern is skewed from 
this by the variation of the sine factor in R. 



-6- 



For short beam path lengths (small n) , the principal and 
subsidiary radiation lobes are broad with maxima strongly 
influenced by the (sine) 2 factor. As n increases, all the 
radiation lobes increase in intensity with the principal lobe 
increasing most rapidly; concomitantly the lobes are displaced 
less from the maxima of (I(u)) 2 . In the limit of infinite beam 
path length, the only significant radiation arises from the 
principal lobe, which is centered on the Cerenkov angle. 

In the regime where the radiation wavelength is much greater 
than the extent of the charge bunches, the variation of R with 
F(k) can be neglected. In this case the position of the maxima 
can be found from the solution of the transcendental equation 



tan u 
u 



- (1 - 



TTn ( sinG ) 2 



) 



(10) 



where cos9 is obtained from (7) 



cos G 



(11 



nS tth 

Equation (10) is expressed as an equation in u by substituting 
(11). After the value of u satisfying (10) is determined, the 
angular position of the maxima is found from (11). Then, the 
maximum value :f the radiation function is found From (4). 

Observation of the radiation patterns is difficult because 
:sf unwanted reflections from :he ground plane ana the walls of 
the experimental chamber. However, taking pains to eliminate 
extraneous signals leads zo reasonably good agreement^ between 
theory and experiment. 



-7- 



Fig. 1, similar to Fig. 4 and Fig. 5 of Reference 2, shows 
the fundamental (n-10) and first harmonic radiation patterns 
( n-20 ) calculated for an electron beam issuing from an S-band 100 
Mev linac into air (n = 1.000268) where it travels 105 cm. 
The effects of changes in the beam length parameter , n, described 
above, are clearly evident in the figure. Fig. 2 calculated for 
larger values of n shows a continuation of the trend of the major 
radiation peak to narrow and grow in intensity as the beam length 
increases . 

Although the above discussion is for the Cerenkov regime of 
radiation (nB > 1), there is no such restricting condition in the 
development leading to (1) and (2), and therefore these equations 
and their consequences are expected to hold for all possible 
values of nB. This leads to an apparent contradiction since it 
is well Known r'rom other calculations ana experiments that in the 
sub-Cerenkov regime (n8 < 1) the strength of the radiation does 
not depend on the path length of the charged particle beam. This 
point is addressed in a later section. 



SHARPNESS OF THE MAIN CERENKOV RADIATION LOBE 
It is difficult to deal analytically with the maxima of W 
even if F(k) has a relatively simple form. But regardless of the 
exact shift of the maxima, the diffraction function always has 
zeroes at u = mtr . The corresponding 9 values are given by 

cose = — ±— - -—- (12) 

m np n 

where m is an integer. For m = ±1 , these limits restrict the 

value of the principal peak of W to lie between the 9 values 

determined by these zeroes in I(u); assuming that these values of 

u correspond to physical values of 9. Substituting m = ±1 into 

(12) gives 

cos9 a = (nB)" 1 + n _1 (m--1) (13) 

cos9 b = (n6) _1 - n" 1 (m=+1 ) (14) 

for uhe upper ( 9^ ) and lower (9 a ) bounds of the main peak. 

The behavior of the main radiation lobe, bounded by the 
angles 9^ and 9 a , depends on the constants nB and n. It is 
obvious ohat as n + » , ohe lobe narrows and both 9^ and 9 a 
approach 9 C , assuming, of course, that n8 > 1 and 9 C is defined. 
In this Limit Df an infinite medium, the radiation all appears at 
cne Carenkov angle. 

In che other extreme, as n becomes smaller, diffraction 
spreads out the main lobe, and 9^ increases from 9 C to eventually 
become 180° for the value n^ of the beam length parameter, where 

nb=n6(n6+1)" 1 (15) 

For realizable values of n$, nt> has a value of approximately 1/2. 



-9- 



Similarly, as n decreases, 9 a diminishes and becomes zero 
for n = n a , where 



Ti a = n8(nB-1)~ 1 



(16) 



One notes that n a is larger than nt» , and that n a varies 
considerably depending on the value of n0. For realizable values 
of n&, n a approaches 1; wheras for n$ only slightly greater than 
one, n a is quite large. For example, 100 MeV electrons in air 
(n& = 1.000255) have an n a value of 3920 while the same electrons 
in water (n8=1 .333) have an n a value of 4. 

For path lengths shorter than n a only the upper bound has 
physical reality. This does not mean that a Cerenkov radiation 
peak does not occur for these short beam lengths, but only that 
the peak bound suggested by (13) is inapplicable and that the 
lower bound on the peak angle is zero. 

3ehaviour of the two angular bounds is shewn in Fig. 3 for 
100 MeV electron bunches from an S-band Linac propagating in air 
( 9 G = 1.3°) ana water ( 8 C = 41.4°). For both materials, the 
angular difference (9^ - 9 a ) is large for relatively short beam 
paths but as n increases, the difference diminishes and both 
radiation patterns approach a o like function centered about 9 C . 

As mentioned earlier, the main radiation peaK is sensitive 
to ".he form factor so ~hat it is iifficult to determine 9 m , the 
value of 9 for which the radiated power is a maximum, except by 
numerical studies. Fig. 3 also shows such numerical results for 
air, obtained from the calculations which led to Fig. 7 of Ref . 
2. Taking the lower bound to be zero when 9 a does not exist, the 
graph shows that as a rule of thumb, 9 m occurs roughly midway 



10- 



between the bounds 9^ and 6 a . As pointed out in Ref . 2, the 
spreading of the main lobe of radiation about 9 C is assymetric 
from the sine factor in (4) so that 9 ra is larger than e c . 



-11- 



EMISSION THRESHOLD 

The above discussion showed that as n varies, the upper and 
lower bounds and therefore the peak between them can change 
position, and Fig. 3 shows the effects of varying path length at 
constant electron beam energy, i.e. as n increases, 65 and 9 a , 
both move toward 9 C . 

Both the beam length and the beam energy (through B) affect 
the position of 65 and 9 a . At some finite n the radiation 
pattern is spread into a diffraction lobe bounded by 9^ and 9 a . 
As the beam energy, and thus B, is reduced, 9^, 9 C , and 9 a become 
smaller. The angles may become non-physical because the 
governing equations contain cos 9 which formally may exceed unity 
Since the inequality 9 a < 9 C < 9^ is always satisfied, it is 
possible to have only 9 a be non-physical as discussed in the 



previous section, or go have both 9 a and d c non-pnysical 



in 



either case, the resulting main lobe of radiation extends from 
zero degrees to 9^ and this phenomenon may be termed sub-Cerenkov 
radiation because it occurs for nB less than (but usually close 
to) unity. More precise delineation of parameter ranges for nB 
and n are discussed below. The spreading of the lobe and the 
conditions for sub- threshold Cerenkov radiation depend only on 
the parameter n, which depends on L and A. The beam bunch size 
parameters enter only because long wave length Cerenkov radiation 
is strong only for bunched beams. 

We define the onset or threshold of the emission of Cerenkov 
radiation to be the situation when 9^ begins to enter the 



-12- 



physical range. Then setting 9^=0 in (14) gives the threshold 
relation. 

nB = n (n+1 )~ 1 (17) 

A plot of (17) is shown in Fig. 4. As the path length 
increases, the product nB first rises rapidly and then 
asympotically approaches the value unity. For values of n6 > 1 , 
the Cerenkov angle 9 C is in the physical range and diffracted 
Cerenkov radiation is emitted. For values of nB and n between 
the curve and unity, 6 C is nonphysical but radiation with a well 
defined peak is still produced. For values of nB and n below 
the curve, radiation is emitted with a rapidly oscillating 
spatial dependence. Although Fig. 4 is a universal curve, it is 
useful and instructive to construct threshold energy curves for 
particular materials. 

Using the usual relation between B and Y, (17) can be 
written in terms of Y^ , the value of Y necessary for the onset of 
sub-Cerenkov radiation. 



Y t (n) = [1 - 



n 2 (1 + n" 1 ) 2 



-1/2 



(18) 



This gives the required energy E^ = v - E for the onset of 
emission in terms of the index of refraction n and the path 
length n. The energy required for onset of emission is then 
given by E t = Y t E . 

Limiting values of (18) can be obtained for very long and 
very short path lengths. For infinite path length 

-13- 






y (n--) - [1 - -V]" 1/2 



(19) 



which is the same condition as nS = 1, the usual threshold for 
emission of Cerenkov radiaton. For large n, Tt(n = °°) approaches 
the value of 1 . If n-1 as for most gases, the threshold value of 
Tt(n = co ) is large and depends critically on the particular value 
of n. Then, introducing the refractivity 5 and writing the index 
of refraction as n=1+5, the threshold value of Yt(n = °°) is 
proportional to 6" 1 / 2 ; and in the limit of small 6 



Y t (n — ) = (26)"1/2 



(20) 



From (20), the threshold energy at infinite beam length is 22.1 
MeV for electrons in air and 60.2 MeV for electrons in helium. 

For short path lengths, (18) shows that, as n -*■ 0, 
Yt(n=0) •*■ 1 independent of the value of n. Thus for very short 
paths, there is no threshold. This may be seen from (17) where 
as n + , the value of 3 at threshold also approaches zero. 

Since many charged particle beams are composed of electrons, 
it is convenient to display threshold energy (instead of Y^) as a 
function of beam length as is shown in Fig. 5. Plots for three 
materials with different indices are shown: all approach 0.511 
MeV :ov snort path lengths and approach the value given by (19) 
for long beam lengths. 

From (19), the variation of the threshold emission energy of 
a medium with a large index of refraction is small. Thus for 
water, the emission threshold is relatively independent of 
particle beam path length, varying between 0.511 and .077 MeV. 



-14- 



For gases the variation is larger - over two decades in the case 
of helium. 

This large variation in threshold energy with path length 
means that Cerenkov-like radiation can be produced by short beams 
with energies substantially below the threshold energy for 
infinite path length. For example, a 10 MeV electron beam with a 
length of n = 10 would be well above the threshold for either 
helium or air, but would be far below the infinite path length 
threshold values of 60.2 and 22.1 MeV. Such a beam would produce 
Cerenkov-like radiation in either medium. 

Since the thresholds for the two gases are different for the 
larger path lengths, it is possible to find sets of parameters 
where one gas is favored. A beam with an energy of 18 MeV and a 
length parameter n - 4x103 would produce Cerenkov-like radiation 
*men propagating in air ouc nou nelium. 

Fig. 3 shows how the width of the main lobe varies with the 
path length of 100 MeV electron bunches. For other beam energies 
above Et(n= ao ), the threshold curves for infinite path length are 
similar except displaced. As the beam energy decreases the 
Cerenkov angle C which is the asymptote of the Ovj and Q a curves, 
is lowered, and consequently it ts approached at increasingly 
larger path lengths. For beam energies very close to 
£ t= Y+. ( n = °°) E the asymptotic nature is not evident until extremely 
long path lengths are attained. (This behaviour is not 
surprising since at E t the Cerenkov angle is zero at infinite 
path lengths ) . 



-15- 



The limits on path length for either Q a or 0^ to be physical 
are obtained from (13) and (14) by setting the angle equal to 
zero. The limiting path length for 9 a to be nonphysical is n a as 
given by (16), and the approach of a to this limit for beam 
energies well above E^ is shown in Fig. 3- For 65 , the limiting 
value of path length denoted n L » is 

n L = n6 (1-n6)" 1 (21 ) 

which gives a non realistic (negative) value for beam energies 
above E^ and a positive value for energies below E^. 

Consequently the behaviour of curves like those in Fig. 3 is 
different for energies less than E^. For these energies, (16) 
gives a negative result for n a and therefore for these energies 
only 0t> is physical, and only for path lengths less than nj.. 



-16- 



ENVELOPE OF THE RADIATION 
The diffraction analysis of the radiation is correct in 
predicting the angular dependence and intensity of the radiation, 
and its variations with frequency, and path length of the charged 
particle beam. However, an understanding of the sub-Cerenkov 
radiation patterns and their development into the characteristic 
Cerenkov shape is more easily displayed with a different 
expression for the previous formulation. 

Substituting for u in the diffraction function allows the 
radiation function to be written 



R = 2 F(k) sin u G(n6, 6) (22) 

where G(nB,9) is a function that often arises in radiation 
calculations , 



G(n8,9) =- 



sine 



(n8) ' -cos 



(23) 



Both the radiated power W and the radiated energy E are 
proportional to R^, and the radiation patterns can be thought 
to be an oscillatory function modulated by an envelope. If 
variation of R with F(k > is neglected, then aside from some 
constants, (G(n8,9))2 is the envelope of the oscillating 
sin^u function which takes on values between zero and one. The 
form of the envelope depends upon the value of n& . Either there 
is a peak at cos9 s =nB, or there is a pole at cos9 c =1/nB; only one 
possibility is allowed. Regardless of the value of n8, the 
function has zeroes at 9=0 and 9 = tt. 



-17- 



Radiation is in the sub-Cerenkov regime when nB<1 . Then the 

envelope function has a peak at 9 S (cos9 s =n8) of height equal to 

(cot2e s ). In this case, the height of the peak does not depend 

upon the path length of the charged particle beam. Consequently, 

the largest possible value of the radiation function in the 

sub-Cerenkov case is 

-*- 
R2(e s ) „ 4 F 2 (k) cot 2 e s (24) 

regardless of the length of the beam. 

As nB increases, the peak angle 9 S decreases and the height 
of the peak grows. When nS=1 , the peak angle goes to zero, the 
envelope function has a pole (at 9=0), and the Cerenkov regime is 
attained . 

In the Cerenkov regime (nB>1 ) the envelope has a pole at the 
Cerenkov angle (cos 9 c = 1/nS). However, since sin u is always 
zero at the Cerenkov angle, the radiation function remains 
finite, with a value of 

R2(e c ) = 4 7T 2 n 2 F 2( k ) si n 29 c (25) 

In the Cerenkov case, the height of the peak depends explicitly 
upon the value of the beam length. 

Fig. 6 shows a plot of the envelope function for the two 
cases. For illustrative purposes the curves are drawn for a plus 
and minus two per cent variation from the Cerenkov threshold of 
nB = 1 . These values of n?> give 83 = 11.48 degrees and 9 C = 11 - 36 
degrees . 

Since (7) shows that u is a function of 8, as 9 varies from 
zero to tt, u will go through many cycles, the exact number 



-18- 



depending on the value of the path length parameter n. 
Successive zeroes of sin^u will occur at integral multiples of it, 
with maximum values (of 1.0) occurring halfway in between. The 
radiation pattern, proportional to the product of sin 2 u and 
(G(nB,9)) 2 then will have zeroes and maxima at the corresponding 
theta values. 

In the limit of very long beam path length, the sin 2 u 
function oscillates very rapidly causing the adjacent peaks to be 
very closely spaced in 6. This behavior can be seen by 
differentiating (7) to obtain 

6u = Trn sine 69 (26) 

The peaks of (sin 2 u) occur at intervals of tt, and setting 
6u = tt gives 

59=(nsin0)- 1 (27) 

showing that the peak spacing becomes very small as n increases. 



-19- 






SUB CERENKOV RADIATION PATTERNS 

To illustrate the development of the radiation patterns, 
consider the pattern formed with the sub-Cerenkov envelope in Fig 
6, drawn for nB=0.98, corresponding to 2.55 MeV electrons 
traveling in air. Fig 7 shows the radiation patterns in the 
forward direction calculated for ng=0.98 and two different values 
of n. The envelope of the pattern is clearly evident for angles 
greater than approximately 20 degrees. Since the spacing of the 
oscillations is greater at lower angles the envelope is less 
evident at these angles. As the radiation angle increases to 
values larger than shown, both curves continue to have a 
monotonically decreasing set of peaks. 

For n=12 the radiation has a single dominant peak and 
several subsidiary ones. For n=37 the largest peak is not the 
first, and there are several secondary peaKs of approximately the 
same height; no single one being outstanding. If the beam path 
length is increased still farther, many closely spaced peaks 
appear inside the envelope, with no single peak being dominant. 
The radiation pattern begins to be rapidly oscillating as shown 
in Fig 8 for n-1 00. 

Consequently, in the sub-Cerenkov case there are two types 
of radiation patterns; those without a single dominant peak, and 
those that have a dominant peak giving the appearance of 
diffracted Cerenkcv radiation. The latter case we call 
psuedo-Cerenkov radiation.- 

The difference between sub-Cerenkov radiation patterns with 
a dominant peak and those with peaks of roughly the same size is 



-20- 



one of path length. When the oscillating sin 2 u function has a 
maxima near the maximum of the envelope function, that peak will 
be largest. The positions of the maxima in sin 2 u are found by- 
setting u in (7) to be an odd multiple of tt/2, giving 

J 2 



cose = 



(28) 



'p nB 2n 
where p is an odd integer. If these maxima are to occur at the 

peak of the envelope function, the angle given by (28) is equal 

to the peak angle 9 S . Substituting cos9 s =n8 in (28) yields the 

condition that successsive maxima will occur at S 



1 - (ng)2 



= sin0 s tan6 s 



(29) 



2n n6 

which shows that if p is incremented, n also must change in order 
to maintain a maxima at 6 S . 

Fig 7 is calculated for p=1 (solid line) and p = 3 (dashed 
line). The solid line shows a single dominant peak with a ratio 
of the height of the first to second peak of nearly two. 
This curve appears similar to the Cerenkov radiation patterns 
shown in Fig 1 , and therefore is an example of psuedo-Cerenkov 
radiation. Since the lobes of the dashed curve are more closely 
spaced, this curve has a lower ratio of first to second peak 
heights and this pattern has begun to approach the rapidly 
oscillating behaviour region of Fig 8. For large values of p 
(and therefore n) the distinction of a maxima occurring at 8 S 
becomes less important, and these patterns become practically 
indistinguishable 'from patterns like that in Fig 8 which was 
calculated for an arbitrary value of n. 

-21- 



CERENKOV RADIATION PATTERNS 
In the Cerenkov regime, development of the radiation pattern 
is calculated by methods similar to those employed in the 
previous section. The oscillating sin^u function is again 
modulated by the envelope function, Eq (23) which now has a 
singularity at 9c. The position of the maxima of their product 
is displaced from 9 C and with a value different from (25). 
Because the envelope function is so strongly peaked in this case, 
there is always a single portion of the sin^u function that is 
greatly enhanced by the envelope function. This part of the 
radiation pattern gives rise to the main Cerenkov radiation lobe 
which is usually the one nearest to the direction of travel of 
the charged particle beam. As n increases the envelope function 
narrows about the Cerenkov angle and increases in height with 
the result that a single radiation lobe becomes increasingly 
dominant. In the limit of infinite path length, the radiation 
has only a single delta function like lobe centered at the 
Cerenkov angle. 

Fig 1 and Fig 2 show radiation patterns for several 
different path lengths for 100 MeV electrons propagating in air. 
Taking the microwave value of the index of refraction to be 
n=1. 000268 gives nB=1. 000255, placing the peak of the envelope 
function at the Cerenkov angle of 3 G =1.3°. Fig 1 is calculated 
for n=10 and 20; Fig 2 for n=150, 250 and 1000. All the curves 
show a main peak and one or more subsidiary peaks, and the 
narrowing and growth in intensity with increasing path length 
described above . 

-22- 



In the Cerenkov case, sin^u is identically zero at the 
Cerenkov angle. Other zeroes occur when u = miT, where m is an 
integer. The set of angles at which sin u vanishes has already 
been given by (12) with the previously defined angles of 
9b , 9 C , and 9 a corresponding to m=+1 , 0, and -1 respectively. 
The set continues with positive and negative m values for which 
| cos6 m | S1 . 

In order to have a small maximum precede the principle 
radiation lobe, another angle at which sin u vanishes must exist 
between 9=0 and 9=9 a . That is, the angle corresponding to a 
further negative m value must become physical. Setting 9 m =0 
gives a condition on the path length for addtional minor maxima. 

n - m — ^ (30) 

1 - n8 

For n8>1 , the denominator in (30) is negative which in 
combination with a negative m value gives a positive result 
for n. 

The next negative value is m=-2 so that (30) with that value 
of m should give an n value leading to a minor radiation lobe 
preceding the major one. Using nS= 1 .02 and m=-2 in (30) gives a 
path length of n=102 at tfhich the minor lobe between ^=0 and 3 C 
is evident . 

Fig 9 is a plot of the radiation distribution for 100 MeV 
electrons traveling in a medium for which the index of refraction 
is n=1. 020013- This unrealistically large index for a gas gives 
n0=1.O2 as in Fig 6-8. The solid curve, calculated for n-102, 
shows a small peak preceding the principal one with a ratio of 

-23- 



peak heights of - 66. The dotted line, calculated for a beam 
length half as long (n=51), does not show such a peak. 

For the same electrons in air, the lower index of air 

requires a longer path length of n=7846 at which the minor peak 

becomes apparent. Calculations confirm this, as well as showini 

other minor peaks when the path lengths found by substituting 

larger negative m values into (30) are attained. 



-24- 



DISCUSSION 

The body of this paper is concerned with the production of 
radiation by a charged particle beam in which both the path 
length and the size of an individual charge bunch are finite. 
The effects of the size and shape of an individual charge bunch 
enter only through a multiplicative form factor and do not cause 
the radiation to spread in angle, or modify the threshold. Thus 
our main point is that the beam path length parameter n affects 
the production of radiation by a charged particle beam. The 
spatial pattern of the radiation, its intensity, and the beam 
energy necessary for producing Cerenkov or psuedo-Cerenkov 
radiation are dependent on n. The complimentary diffraction and 
envelope forms both describe these effects, with the latter 
giving a more direct description of the development of the 
radiation patterns in the sub-Cerenkov case. 

Although nS determines the position of the Cerenkov angle, 
the value of n controls the broadening of all the radiation lobes 
Since the peak of the main lobe occurs in the vicinity of the 
peak of the Cerenkov envelope its height also depends on n. In 
the sub-Cerenkov case, n8 determines the position and height of 
the envelope and "he change from psuedo-Cerenkov radiation to a 
rapidly oscillating one is dependent on n. 

A distinguishing feature of Cerenkov radiation is the 
dramatic increase of intensity of the main radiation lobe with an 
increase in beam length as shown in Fig. 1, Fig. 2, and Fig. 9. 
In these figures, the ratios of the peak heights is very nearly 
equal to the square of the ratio of beam length parameters, after 

-25- 



the different positions of the peaks is taken into account. In 
contrast, in the sub-Cerenkov regime, the height of the radiation 
peaks is bounded by the finite height of the envelope function as 
shown in Fig. 7 and Fig. 8. 

In a medium with a large index of refraction, such as water, 
the main radiation lobe occurs at a rather large angle to the 
beam and the lobe broadening subsides quickly for n values 
greater than 20. In addition, the dependence of the threshold 
energy on n is small. Contrarily, in a medium with an index 
close to one, such as a gas, the effect of n is more pronounced. 
The lobe broadening persists in variation of n over several 
decades before approaching the (much smaller) infinite path 
length limit of 9 C . Accompanying this behaviour, the threshold 
energy also varies over several decades of n so that the onset of 
radiation as a function of beam energy is not a sudden phenomena. 

These effects are most apparent at small values of n (-10) 
and are of interest when observing microwave radiation produced 
from an RF linac. Although a high energy, high intensity charged 
particle beam may be shielded by a plasma sheath, Cerenkov 
radiation and the associated broadening of the main lobe as a 
result of a short beam length may arise from the regions at the 
ends of the beam. 

Small radiation maxima may exist closer to the direction of 
travel of a charged particle beam than the major lobe. In the 
sub-Cerenkov case., the minor maxima occur when n increases so 
that the psuedo-Cerenkov regime goes over to the rapidly 
oscillating one. Appearance of the minor maxima depends on n8 as 

-26- 



well as n. For gases, n is slightly larger than 1, and an upper 
limit for is also 1 so that a realistic upper limit is nB-1.001 
This limit leads to e c =2.6°, and n=2000 in order to have a minor 
maxima precede the main lobe; values not very different from 
those found for 100 MeV electrons in air. In a denser medium 
the situation is different. In water where 9 C =41°, a value of 
n=8 would be sufficient to ensure that a leading minor maxima 
occur. Larger n values would have the possibility of producing a 
Cerenkov radiation pattern with many minor maxima preceding the 
main one. 

In media with index of refraction near 1, the radiation 
described here has the same spatial characteristics as transition 
radiation . If the index is set equal to 1 , the peak of the 
radiation envelope occurs at cos 9 S = 6. Then sin 9 S = Y~' 1 , or for 
small 8 S , 9 S =Y~ : wnich xo a characteristic of transition 
radiation . 

As a final remark, it might be argued that the emission 
occuring below the usual Cerenkov threshold should not be named 
sub or pseudo-Cerenkov . But the argument in favor of retaining 
this name is that the radiation has a similar spatial appearance, 
the same polarization as Cerenkov radiation, and the- transition 
to Cerenkov radiation as the energy is increased is smooth and 
cont i nuous . 



-27- 



ACKNOWLEDGEMENT 

This work was supported by the Naval Sea Systems Command 
and the Defense Advanced Research Projects Agency. 



-28- 



REFERENCES 

1. F. R. Buskirk and J. R. Neighbours, Phys Rev A28 , 1531-38 
(1983). 

2. John R. Neighbours, Fred R. Buskirk and A. Saglam, Phys Rev 
A29 , 3246-52 (1984). 

3. X. K. Maruyama, J. R. Neighbours, F. R. Buskirk, D. D. 
Snyder, M. Vujaklija, and R. G. Bruce, J Appl Phys 6_0(2) 518 
(1986). 

4. J. R. Neighbours and F. R. Buskirk, US Naval Postgraduate 
School Report No. NPS 61-84-010, 1984 (unpublished). 

5. Fred R. Buskirk and John R. Neighbours, US Naval Postgraduate 
School Report No. NPS 61-84-007, 1984 (unpublished). 

6. A. P. Kobzev, Yad. Fiz. 27, 1256 (1978) [Sov. J. Nucl. Phys. 
27, 664 (1978) ]. 

7. A. P. Kobzev and I. M. Frank, Yad. Fiz. 3J_, 1253 (1980) [Sov. 
J. Nucl. Phys. 31, 647 (1980)], and Yad. Fig. 34, 125 (1981) 
[Sov. J. Nucl. Phys 34, 71 (1981)]. 

8. A. Bodek, et al , Z. Physics C 18, 299 (1983) 



-29- 



9. X. K. Maruyama, J. R. Neighbours and F. R. Buskirk, IEEE 
Transactions on Nuclear Science NS-32, 199^ (1985). 



-30- 



FIGURE CAPTIONS 
Fig 1 . Calculated radiation intensity as a function of angle for 
an electron beam with a path length of 105 cm issuing from an 
S-band, 100 MeV linac into air (n = 1.000268). The solid curve 
is the pattern for radiation emitted at the fundamental frequency 
of 2.85 GHz so that the beam length parameter n has a value of 10 
The dashed curve is the pattern for radiation emitted at the 
first harmonic frequency (n=20). 

Fig 2. Calculated radiation intensity as a function of angle for 
an electron beam issuing from an S-band, 100 MeV linac for longer 
beam path lengths. The dotted, dashed and solid curves are for 
electron beam path lengths of 1574 cm, 2624 cm and 10496 cm 
respectively. The corresponding n values are 150, 250, and 1000. 

Fig 3- First diffraction lobe angular limits 0^ and Q a as a 
function of beam length parameter n for 100 MeV electron bunches 
traveling in air and water (n=1 . 333) • The lower limit 9 a goes to 
zero at the beam length parameter value na . The dashed curve 
marked 9 m is the calculated angular value at which the peak of 
the main lobe occurs. (Values were obtained from the 
calculations leading to Fig 7 of Reference 2). 

Fig 4. Threshold value of ng as a function of beam length n. 
Values of ng>1 give rise to Cerenkov for all values of n. A. 
value of nB<1, but above the curve gives rise to pseudo-Cerenkov 
radiation . 

-31- 



Fig 5. Threshold electron bunch energies as a function of n for 
water (n-1. 333), air (n-1. 000268) and helium (n=1 . 000036) . At 
large values of n each curve approaches its respective E?. 

Fig 6. Envelope function (G(ng,e)) 2 as a function of angle. The 
solid curve (Cerenkov) is for n$»1.02. The dashed curve 
(sub-Cerenkov) is for n3=0.98. 

Fig 7. Sub-Cerenkov radiation patterns calculated for ng=0.98. 
The solid curve, calculated for n-1 2 is an example of 
psudeo-Cerenkov radiation. The dotted curve, calculated for n=37 
shows a minor maxima preceding the largest lobe and a maxima of 
nearly the same intensity following it. 

Fig 8. Hapidly oscillating sub-Cerenkov radiation pattern 
calculated for nS=.98 and n-1 00. Two minor maxima precede the 
largest lobe. 

Fig 9. Cerenkov radiation patterns calculated for 100 MeV 
electron bunches traveling in a medium with an index of 
refraction of n-1. 020013 and thus giving n$-1.02. The dotted 
curve is for n-51 . The solid curve for n = 1 02 snows a minor 
maxima preceding the principal radiation lobe. 



■32- 



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-33- 



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-34- 



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-35- 



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-41- 



DISTRIBUTION LIST 



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