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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 Approved for public release; distribution unlimited Prepared for: Defense Advanced Research Projects Agency DARPA/STO L400 Wilson Blvd. FedDocs D 208.14/2 „ ,^ ' nnt , Arlington, VA 22209 NPS-61-87-004 NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral R. C. Austin D - A - f hrady . . Provost Superintendent The work reported herein was supported by the Defense Advanced Research Projects Agency. Reproduction of all or part of this report is authorized. This report was prepared by: Unclassified JRITY CLASSIFICATION OF THIS PAGE DUDLEY KNOX LIBRARY REPORT DOCUMENTATION PAGE S^' g^^^L "report security classification Jnclassified lb RESTRICTIVE MARKINGS ■SECURITY CLASSIFICATION AUTHORITY 3. DISTRIBUTION /AVAILABILITY OF REPORT RECLASSIFICATION / DOWNGRADING SCHEDULE IRFORMING ORGANIZATION REPORT NUMBER(S) IPS .61-87-004 5. MONITORING ORGANIZATION REPORT NUMBER(S) iJAME OF PERFORMING ORGANIZATION aval Postgraduate School 6b. OFFICE SYMBOL (If applicable) 7a. NAME OF MONITORING ORGANIZATION ..DDRESS (City, State, and ZIP Code) onterey, CA 93943 7b. ADDRESS (City, State, and ZIP Code) j.IAME OF FUNDING /SPONSORING rganization Project Agency jlense Advanced Research 8b. OFFICE SYMBOL (If applicable) 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER N6227186WR60058 DDRESS (City, State, and ZIP Code) ^PA/STO 13 Wilson Blvd., Arlington, VA 22209 10. SOURCE OF FUNDING NUM8ERS PROGRAM ELEMENT NO. 62714E PROJECT NO. 6A10 TASK NO 5371/1 WORK UNIT ACCESSION NO. ITLE (Include Security Classification) Cerenkov and Sub-Cerenkov Radiation from a Charged Particle Beam ERSONAL AUTHOR(S) 1st page :. 13b. "iME COVERED Il4 DATE OF REPORT ( Year, Month, Day) ]lS D AGE COUNT -ROM Aug. 36to Mar 37 | March 3, 1987 J TYPE CF REPORT mical JPPLEMEN TARY NOTATION iOSATI CODES '8. )U8J£CT "ERMS Continue on reverse if necessary ana .cenr/ry ov jiock numoen IELD GROUP SU8-GROUP 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. STRI8UT:ON/ AVAILABILITY OF ABSTRACT EjjNCLASSIFIED/UNLIMITED D SAME AS RPT. □ OTIC USERS 21. ABSTRAC SECURITY CLASSIFICATION AME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE (Include Area Code) tic. OFFICE SYMBOL )RM 1473, 84 mar 83 APR edition may be used until exnausted. All other editions are obsolete SECURITY CLASSIFICATION OF THIS PAGE ft U.S. Government Printing Office: 1986—606-24. 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- 0.04 U 0.03 - o.oe - 0.01 - 15 30 ANGLE (cleg) 45 -33- 3.0 U 2.0 - 1.0 - 0.0 2.5 5.0 ANGLE (deg) 7.5 -34- ±1111 II 1 UN 1 1 1 1 1 I him i i i i - ~ /I g/l © / j / 1 \ o E - i i nun UATER / ' / 1 / / / / - = / / / / / / / / ° E / 7 E — *** 1 © I \ / f - — / — — / - ifl 1 1 11 ! IIMI J 1 1 ! in i i i i i i CD CD CO CD CO CD CO 1 CO CO £ e> z: Ul _j 21 CO CO CO CO CO CO CD S3 CO CO CO CO <x z: ta _j u -35- 1.2 nf 0.8 - Q.S - 0.4 - 0,2 - 0.0 0.1 1 10 10 2 10 3 1} -36- S3 S3 U3 CD ~ UJ — (S3 si UJ S3 5) S3 S3 S3 S) S3 CO S) Ul S3 _J S3 — ••— E — ' CC __l UJ - S3 CQ = S3 = S3 — - — 1 - S3 - S3 ] 11 1 I I 1 [ I SI — UJ ac r ~3 >■ -37- 100 50 - 20 40 ANGLE (deg) C -38- 0.010 (J 0.005- 15 30 ANGLE (deg) 45 -39- 0.010 u 0.005- 15 30 ANGLE (deg) 45 -40- 1.5 U 1.0 0.5 10 20 AtfGLE (deg) 30 -41- DISTRIBUTION LIST CDR William Bassett 1 PMS 405 Strategic Systems Project Office Naval Sea Systems Command Washington, D.C. 20376 Dr. Richard Briggs 2 L-321 Lawrence Livermore National Laboratory Box 808 Livermore, CA 94550 F. R. Buskirk & J. R. Neighbours 20 Naval Postgraduate School Physics Department, Code 61 Monterey, CA 93943 The Charles Stark Draper Laboratory 1 ATTN: Dr. Edwin Olsson 555 Technology Square Cambridge, MA 02139 Dr. W. 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