Microwave Cherenkov radiation as a diffraction phenomenon

Nuclear Instruments and Methods in Physics Research B24/25 (1987) 921-924 North-Holland, Amsterdam

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M I C R O W A V E C H E R E N K O V R A D I A T I O N AS A D I F F R A C T I O N P H E N O M E N O N Xavier K. M A R U Y A M A National Bureau of Standards, Gaithersburg, MD 20899, USA F.R. B U S K I R K a n d J.F. N E I G H B O U R S Department of Physics, Nat'al Postgraduate School, Montere), CA 93943, USA

Cherenkov radiation results when the velocity of a charge particle exceeds the phase velocity of light in a mcdium. The wave front moves in a direction given by the Cherenkov angle 0c with respect to the dircction of the moving charge. The appearance of the well defined Cherenkov cone is a consequence of the assumption of an interaction length which is infinite when compared to the wavelength of the radiation observed. When the interaction region of the charged particle with the medium is finite, the Cherenkov radiation pattern is modified by diffraction. This condition is readily attained for microwave wavelengths. Microwave Cherenkov radiation in air has been obse~'ed to exhibit this diffraction characteristic. Coherence of radiation emitted from all the electrons in a mlcropulse of an rf linac has also been observed. For electron bunches periodic in time with frequency %, radiation is emitted at vo and harmonics thereof, in contrast to the continuous frequency distribution observed for a single charge. These and additional consequences of a finite interaction length are discussed.

1. Introduction The study of Cherenkov radiation has" traditionally emphasized radiation emitted in the optical region [1-4]. Cherenkov radiation is characterized as occurring when the velocity of a charged particle exceeds the phase velocity of light in the medium and is observed as a Mach front of radiation emitted at a specific Cherenkov cone angle. This radiation has a continuous spectrum. The availability of intense relativistic electron beams has provoked our interest in the study of Cherenkov radiation at frequencies other than the optical. Although the diffractive feature of Chercnkov radiation has appeared indirectly in classical derivations [1-4] these diffraction characteristics have not been fully apprcciated because any macroscopic dimension is effcctivcly infinite for optical wavelengths. The main optical effect was reported by Kobzev and Frank [5], who observed Cherenkov radiation produced in thin mica layers. In a series of articles [6-11], we have explored the consequences of a finite interaction length for Cherenkov radiation. Some of the interesting observations to be made include the following: (a) The traditional Cherenkov cone is broadened in the form of a diffraction pattern with similarities to optical single slit diffraction and to finite endfire antenna arrays. (b) When the source of radiation is periodic, as is the case for electrons accelerated by a traveling wave rf linear accelerator, the Cherenkov radiation is not a continuous spectrum, but rather consists of discrete frequencies corresponding to the frequency of the electron bunches

and its harmonics. (c) When the source size, i.e., the electron bunch dimensions, is less than the Cherenkov radiation wavelength, all the electrons in the bunch radiate coherently. Consequently, the radiation at low frequencies has observable power. (d) The Cherenkov radiation intensity is proportional to the square of the Fourier transform (form factor) of the source charge distribution. (e) In considering the time structure the frequency integrated power depends on d i / d t , i.e., a dc current, such as a uniform cylindrical beam, does not emit Cherenkov radiation. Because of the diffractive nature of Cherenkov radiation from a finite interaction region, the radiation does not abruptly appear at a threshold velocity equal to the speed of light in the medium, but appears at subthreshold velocities as well. Cherenkov radiation is not a different radiation to be added to other forms of radiation when v > c(medium) but appears naturally in a correct calculation of the radiation.

2. Cherenkov radiation from charge distributions The energy per unit solid angle in a frequency interval, d r , radiated by a charge q, traveling at a velocity v greater than the speed of light, c, in a medium has a continuous spectrum. When the interaction region has length L, it is given by [6,12] E ( v ) dv = QR z dr,

(1)

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922 where Q is the constant

Q =#cq2/S~z 2.

(2)

The speed of light in the medium is c = (/Le) -1/2. The radiation function is R = kL(sin 0) I ( u ) F ( k )

(3)

with

u = kL (cos 0c - cos 0),

(4)

I ( u ) = (sin u ) / u ,

(5)

where k is the wave vector of the emitted radiation in the medium (k = w/c). The classical Cherenkov angle is given by cos 0~ = (nil) - t , where n is in the index of refraction, n = (Co/C) and co is the speed of light in vacuo. F(k), the form factor, is the Fourier transform of the charge distribution o(r).

V(k)=l fd3xe x p ( - i k . r ) p ( r ) .

(6)

When the source is periodic with frequency vo and has charge q per bunch, then the power radiated per unit solid angle is given by an expression [6] similar to eq. (1):

IV(v) dv -----v~QR 2 dr.

(7)

In this case there is a restriction on the allowable frequencies so that

k =j(2~rVo/C),

j = 1, 2, 3 ....

j is the harmonic number. In these equations, I(u) describes the diffraction nature of Cherenkov radiation. Note that in the usual limit that kL--, oo, the diffractive term exhibits a Bfunction behavior peaked at 0 = 0c. Furthermore, the factor L212(u) is identical in form to the result obtained for the calculation of Fraunhofer diffraction from a slit across which the phase varies linearly, such as plane waves striking the slit at an angle to the normal. However, the diffraction angle is not the usual one measured from the normal to the radiating line source. Here it is more convenient to measure the diffraction angle 0 from the electron beam line so it is the complement of the usual one. The sin 0 factor arises from cross products used in calculating the Pointing vector, and is just the angular factor in the usual expression for power radiating from a dipole oriented along the electron beam. Thus the expressions for energy and power can be interpreted as the interference of dipole radiators whose phase varies linearly with positions along the beam line. The effects due to the finite size distribution of the charge which is radiating in the medium shows up in the form factors, F(k). In the limit of a point charge F(k) is unity, so for Cherenkov radiation due to an elementary charge particle, the form factor effects have been ignored.

When the wavelengths are of the order of magnitude of the physical dimensions of the charge, then the Ctaerenkov diffraction function is modulated by the form factor. This ease is achieved readily when microwave radiation is observed from electrons of a traveling wave rf linae. The electrons do not appear singly, but as bunches riding on the rf wave. For the case of an induction accelerator, where the bunch is many meters long, the form factor effect could have a considerable effect for the radio frequency Cherenkov radiation.

3. Example and the consequences of a finite interaction length To observe the effect of a finite interaction length on Cherenkov radiation, we used 100 MeV electrons from the linac at the Naval Postgraduate School to create microwave Cherenkov radiation in air [7,10]. The accelerator operates at the S band frequency of 2856 MHz. The electron-air interaction length was restricted to 14 cm. The schematic experimental arrangement is shown in fig. 1. Radiation was observed at frequencies of 2.86, 5.71, 8.57, and 11.42 GHz corresponding to the fundamental and the higher harmonics of the S-band rf linae. Using a frequency filter after the receiving antenna, the absence of radiation at nonharmonic frequencies confirms the discrete nature of Cherenkov radiation from a periodic source. Fig. 2 presents the observed pattern for the third harmonic in the X-band. The traditional Cherenkov cone angle arising from an infinite interaction length is 1.3 o. We observe a broadened peak at about 25 o far in excess of that expected at 0c= c o s - l ( 1 / n f l ) . These

Cerenkov Cone

Antenna-Detector

Fig. 1. Schematic experimental arrangement. Electrons pass from the accelerator into air and generate Cherenkov radiation which is reflected by a metal mirror. A hollow cone of radiation is formed as shown since the radiation is generated only in the region between the beam pipe and the mirror. The antenadetector assembly can be translated on a track (not shown). In this figure, no diffraction effects are shown.

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Fig. 2. Microwave Cherenkov radiation at ~,= 3z,o showing the diffraction nature of the phenomenon. The solid curve represents the measured data and the dotted curve is calculated. The interaction length is L = 14 cm. The data have been normalized to the calculated power per unit solid angle assuming q =1.5×10 -]z C. observations reveal the diffraction nature of Cherenkov radiation. For the conditions of this experiment, the electron bunch length is smaller than the observed wavelengths and the conditions were not ideal to reveal the form factor effects. Although our equipment was not absolutely calibrated and our measurcments have been normalized to calculations for the magnitude of the obser,'ed power, it is to be noted that if the electrons in the microbunch were not radiating coherently, we would not have been sensitive to the Cherenkov radiation. Each microbunch of electrons contained about 107 electrons. Since the charge in the expression for the radiated power appears quadratically, incoherent radiation would appear at power levels 10 .7 lower. The lack of sensitivity of our apparatus and the noise level would have made these observations impossible. To consider the effects of the finite charge distribution, we calculated the effects of Cherenkov radiation from long intense electron bunch such as from a relativistic induction accelerator. If the beam is assumed to have negligible transverse dimensions, then the wave number of importance in the form factor is the projection of the wave number along the beam axis, i.e., k : = k cos 0. This condition results in an enhancement of the radiation perpendicular to the beam axis. Fig. 3 illustrates the Cherenkov radiation pattern which might be expected from a 10 kA beam which has a trapezoidal distribution 30 m long. The peak at fonvard angles is the Chercnkov radiation reflecting the diffraction phe-

0.0

0

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180

Fig. 3.25 MHz wavelength Cherenkov radiation pattern calculated assuming a relativistic 10 kA, 100 ns trapezoidal electron pulse. The ordinate is the energy per solid angle per frcqucncy in units of (J/sr)/Ilz. The enhancement at 90 ° results from the form factor effect.

nomena, but the peak near 90 ° is a manifestation of the maximum of the form factor at 90". This somewhat unexpected result predicts that one should not be surprised to find large radiation signals perpendicular to the direction of the beam. As the finite interaction length causes diffraction so that radiation is produced at angles other than the usual Cherenkov angle, the relationship between Cherenkov and other forms of radiation, namely, what is referred to as electromagnetic pulse (EMP) and ordinary dipole radiation becomes easier to understand. Instead of the usual Fourier analysis of the fields, if we look at the time dependence of the fields [8,11], the formulations reveal that the radiation is associated with d l / d t at the leading and trailing parts of the pulse. A finite charge distribution avoids singularities which arise from assuming a point charge. The Cherenkov radiation (for an infinite path) consists of positive and negative pulses separated by a distance, which is the pulse length. For a semi-infinite interaction length, Cherenkov radiation appears within a cone of angle 0,., whose apex is at the start of the path. An EMP pulse appears outside that cone. For a finite pathlength L, both Cherenkov and EMP appear, the latter dominating as L becomes smaller. For L short and /3 small, the fields have a dependence which is essentially a single pulse of dipole radiation. VII. ACCELERATOR TECt INOLOGY

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Fig. 4. A qualitative representation of field lines for an electron bunch traversing a senli-infinite path. As the position of the observer changes, the pulse form changes as shown in the inset. Chercnkov radiation occurs at the lower right; EMP pulses occur to the upper left.

Fig. 4 is a qualitative representation of field lines for an electron bunch traversing a senti-infinite path. As the position of the observer changes, the pulse form changes as shown in the inset. Cherenkov radiation occurs at the lower right; E M P pulse occur to the upper left. Finally, as a consequence of the relaxation of the phasing condition between the moving charge and the radiated wave for finite beam lengths, the requirement v > c for Cherenkov radiation to occur is relaxed. The important parameter for the threshold is the normalized interaction length r / = L / X . Fig. 5 shows that threshold electron total energy as a function of ~ for the case of He, air and water. In the case of an infinite interaction length, the requirement that ~TB> 1 is preserved. By allowing the interaction length to be finite, features of Cherenkov radiation not generally appreciated in the traditional treatment become readily apparent. By the observations and the examples presented here, we hope to have increased the awareness of these aspects and provoke consideration of the use of Cherenkov radiation for applications in addition to high energy particle detection. The advent of relativistic induction linacs and F E L modulated electron beams provide interesting possibilities for the further exploitation of the Cherenkov effect.

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References Ill J.V. Jelley, Cherenkov Radiation and Its Application (Pergamon, London, 1958). [2] V.P. Zrelov, Cherenkov Radiation in ttigh Energy Physics (Atomizdat, Moscow, 1968) [translation: Israel Program for Scientific Translations, Jerusalem, (1970)]. [3] W.K.H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, 1962). [4] J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1963). [51 A.P. Kobzev, Yad. Fiz. 27 (1978) 1256 [Soy. J. Nucl. Phys. 27 (1978) 664; A.P. Kobzev and I.M. Frank, ibid. 31 (1980) 1253 [31 (1980) 647]; 34 (1981) 125 [34 (1981) 71]. [6] F.R. Buskirk and J.R. Neighbours, Phys. Rev. A28 (1983) 1531. [7] J.R. Neighbours, F.R. Buskirk and A. Saglam, Phys. Rev. A29 (1984) 3246. [8] F.R. Buskirk and J.R. Neighbours, Phys. Rev. A31 (1985) 3750. [9] X.K. Maruyama, J.R. Neighbours and F.R. Buskirk, IEEE Trans. Nucl. Sci. NS-32 (1985) 1994. [10] X.K. Maruyama, J.R. Neighbours, F.R. Buskirk, D.D. Snyder, M. Vujaklija and R~G. Bruce, J. Appl. Phys. 60 (1986) 518. [111 F.R. Buskirk and J.R. Neighbours, Phys. Rev. A34 (1986) 3470. [12] J.R. Neighbours, F.R. Buskirk and X.K. Maruyama, US Naval Postgraduate School Report No. NPS-61-86-014, unpublished.