Magnetic mirror cavities as terahertz radiation sources and a means of quantifying radiation...

Magnetic mirror cavities as terahertz radiation sources and a means ofquantifying radiation friction

Amol R. Holkundkar1,a) and Chris Harvey2,b)

1Department of Physics, Birla Institute of Technology and Science, Pilani, Rajasthan 333031, India2School of Mathematics and Physics, Queen’s University Belfast, Belfast, Northern Ireland, United Kingdom

(Received 3 July 2014; accepted 22 September 2014; published online 8 October 2014)

We propose a radiation source based on a magnetic mirror cavity. Relativistic electrons aresimulated entering the cavity and their trajectories and resulting emission spectra are calculated.The uniformity of the particle orbits is found to result in a frequency comb in terahertz range, theprecise energies of which are tunable by varying the electron’s c-factor. For very high energyparticles, radiation friction causes the spectral harmonics to broaden and we suggest this as apossible way to verify competing classical equations of motion. VC 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4897307]

I. INTRODUCTION

It is a well known and extensively utilized result of elec-trodynamics that an accelerated charge will radiate.1 By con-trolling the particle acceleration with carefully chosenconfigurations of the background electromagnetic field, lightsources can be tailored to specific requirements. This is theprinciple behind many devices, such as free-electron lasers,which use undulators to force an electron to oscillate in acontrolled way, resulting in the emission light of exact fre-quencies.2 It is also the fundamental precept governing morecomplex radiation sources such as higher harmonic genera-tion from laser-atom interactions. Similarly, by firing anelectron through a magnetic cavity one can create a gyro-tron,3 a type of free electron maser used as a millimeterwave heat source which has many applications in industry aswell as being used to heat plasmas for nuclear fusion experi-ments. In particular, the design and development of terahertz(THz) radiation sources is a field of contemporary interestbecause of the vast variety of applications it promises. THzradiation sources are useful in such areas as real time imag-ing,4 biological research,5,6 and security. In view of this thereare various devices are being developed including terahertzsemiconductor-heterostructure lasers,7 quantum cascadelasers,8,9 etc.

Here, we present (to the best of our knowledge) a previ-ously unconsidered setup in this context—a magnetic cavitymirror trap. The device consists of a strong magnetic fieldwith a “basin”-shaped/inverted super-Gaussian profile. Thefield strength increases steeply at the ends of the cavity sothat particles injected into it will be reflected back towardsthe central region if they migrate too closely to the edges. Aswe shall see, the result is that electrons injected into the cav-ity will gyrate in a highly regular manner, producing radia-tion of a very narrow bandwidth. Although the particles willalso acquire a longitudinal drift, they will remain in the cav-ity due to reflection, bouncing back and forth off each end ofthe trap. The primary aim is to generate a tunable, high

quality THz radiation source that can be utilized in many ofthe applications described above. Additionally, we will findthat such a setup could potentially serve as a testbed for clas-sical theories of radiation friction (RF).

II. SIMULATION MODEL

We consider the propagation of a relativistic electron ina magnetic field. The spatial profile of the magnetic field isgiven by

Bz zð Þ ¼ Bm $ Bm $ B0ð Þexp $ z$ 0:9L

0:7L

! "12" #

: (1)

Here, B0 and Bm are the minimum and maximum magneticfield strengths, respectively, and L is the length of the cavity.The magnetic field has a spatial profile closely resembling aninverted super-Gaussian. The field profile and the geometryof our simulation along with the possible experimental setupare illustrated in Fig. 1. The desired magnetic field profilecan be constructed by placing a set of solenoids around thecavity. The current through them can be calculated using theBiot–Savart law such that it will produce the desired mag-netic field as given by Eq. (1).10 The radiation from theinjected particles is detected along the z axis with hdetec beingthe angle of detection. It should be noted that Gauss’s lawfor magnetism demands that r % B ¼ 0 and so, as a result,the field will also be non-zero along the x and y directions.Provided @Bz/@z does not vary much along the radius of thecavity,11 the transverse field components will be

Bx x; zð Þ ¼ $x

2

@Bz

@z; (2)

By y; zð Þ ¼ $y

2

@Bz

@z: (3)

The dynamics of a charged particle in a magnetic field isgoverned by the Lorentz force equations

dp

dt¼ q v& B½ (; (4)

a)[email protected] and [email protected])[email protected]

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PHYSICS OF PLASMAS 21, 103102 (2014)

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v ¼ p=m0

c; (5)

dr

dt¼ v; (6)

where, p, v, r, q, and m0 are relativistic momentum, velocity,coordinate, charge, and mass of the particle, respectively.Furthermore, c is the relativistic factor and B is the appliedmagnetic field. The above equations are solved numericallyusing a standard Boris leapfrog scheme.12 The particle orbitis calculated by substituting Eqs. (1)–(3) into the equationsof motion, and specifying the initial conditions for the injec-tion energy and the injection angle hin (which the trajectorywill make with z axis), then the dynamics of the particle areevolved numerically.

Throughout the rest of this article, we will work indimensionless units defined in terms of typical time andlength scales of the problem. Ideally, one would normalizein terms of the gyration frequency of the electron, but in ourcase this quantity is dependent on the initial c-factor of theparticle. Instead, we normalize in terms of a fundamental fre-quency x0¼ 2pc/k0, corresponding to a wavelength k0 of 1m. We can then define a normalized magnetic field ampli-tude b) qB/mex0, where B is the magnitude of the magneticfield in SI units. Thus b0 and bm are the dimensionless mag-netic field strengths corresponding to B0 and Bm in Eqs.(1)–(3). (This is analogous to the dimensionless measure ofintensity, a0¼ eE/x0mc, used in studies of laser-matter inter-actions, see, for example, Ref. 13.) In these units the dimen-sionless magnetic field b¼ 1 corresponds to a magnetic field

strength of 0.01071 T, the dimensionless unit length, k0x¼ 1,corresponds to 1/2p m and the dimensionless unit time,x0t¼ 1, corresponds to 0.53 ns. Unless otherwise stated,throughout the rest of this article, we will take the cavitylength L to be p/5, i.e., 0.1 m. The particle is injected at anangle hin to the field axis and the subsequent radiation emit-ted by the particle is detected along the cavity axis(hdetec¼ 0), as shown in Fig. 1. In Sec. III, we consider spe-cific parameter values.

Next, we consider the radiation emitted by the acceler-ated charge. The energy radiated per unit solid angle per unitfrequency is given by1

d2I

dx dX¼ e2

4p2c

####ð1

$1

n& n$ bð Þ & _bh i

1$ b % nð Þ2eix t0þR t0ð Þ=c½ (dt0

####2

;

(7)

where n is a unit vector pointing from the particle’s positionto the detector (R) located far away along the z-axis, and band _b are, respectively, the particle’s velocity and accelera-tion. In our dimensionless units, if s¼x/x0 is taken to bethe harmonic of fundamental frequency, then above equationsimplifies to

d2I

dx dX¼####ð1

$1

n& n$ bð Þ & _bh i

1$ b % nð Þ2ei s s0þR s0ð Þ½ (ds0

####2

: (8)

Here s)x0t and we have normalized the intensity by thefactor e2/(4p2c). All the quantities in the above equations areevaluated at the retarded time so one can directly do the inte-gration in some finite limit.

The Larmor radius for a relativistic particle in dimen-sionless units is given by rL¼ p?/b0, where p? is the perpen-dicular component of the momentum. The cyclotronfrequency for a relativistic electron is given by xc¼ eB/(cme) which in our dimensionless parameters can be writtenas x0c ¼ x0b0=c. Finally, the spectrum in units of the cyclo-tron frequency can be obtained via x ¼ n0x0c, which impliesx/x0¼ n0b0/c and so we need to simply divide by b0/c tohave a spectrum in harmonics of the cyclotron frequency(here n0 is the harmonic number).

III. GEOMETRY OF THE PROBLEM

The physics of magnetic mirrors is quite well under-stood and routinely used in plasma confinement machinessuch as tokomaks. The basis behind the working of magneticmirrors, and thus also plasma confinement, is the invarianceof the magnetic moment. As the particle moves from a weakmagnetic field region to a strong field region the perpendicu-lar component of its velocity (v?) must increase to keep themagnetic moment constant. Since the total energy mustremain constant, the parallel component (vk) of the velocitymust necessarily decrease.11 If the field at the end of the cav-ity is strong enough then eventually vk becomes zero and theparticle is reflected back. The pitch angle with which the par-ticle is injected in the low field region also plays very crucial

FIG. 1. A schematic showing the possible experimental setup (top) alongwith the geometry of the simulation (bottom). L is the length of the cavity,b0 and bm represent the magnetic field strength in the center region and atthe ends, respectively. The radiation is detected along the z axis.

103102-2 A. R. Holkundkar and C. Harvey Phys. Plasmas 21, 103102 (2014)


role in determining the trapping of the particle in the mag-netic mirror.

In order to gain some insight on the dynamics of the par-ticle in this type of magnetic field setup, we have presentedthe time evolution of the particle trajectory in Fig. 2. Wehave considered the cases where a 511 keV electron (c0¼ 2)is injected into the cavity at angles hin¼p/6 and p/3. Thevalue for the magnetic field at the center is chosen to beb0¼ 1200 (12.85 T); however, the value at the end is deter-mined by the criteria that the particle should be trappedinside the cavity and for that to happen the magnetic field atthe ends should be at least11

bm ¼ b0= sin2hin: (9)

It can be inferred from Eq. (9) that for smaller injectionangles the required value of bm increases. For hin¼ p/6 wechose bm¼ 4810, which is slightly higher than the thresholdmagnetic field (4800). Similarly for hin¼p/3 we chosebm¼ 1610, which is also slightly higher than the thresholdvalue (1600). The simulation ran for s¼ 10 and the trajecto-ries of the particle for these two cases are presented in Fig. 2.The left column shows the result for hin¼p/6 and the rightcolumn for hin¼p/3. As can be seen from these plots, as weincrease the injection angle the pinching effect at the endsreduces as a consequence of the lower magnetic field (Eq.(9)) and hence the trajectories become more and moreuniform.

Keeping this aspect of the magnetic field setup in mind,we find that it is most efficient to inject the particle perpen-dicularly to the magnetic field, since in this case the mag-netic field strengths at the ends will be comparable to thefield at the center of the cavity, as can be inferred from Eq.(9). We also note that the field strengths along the transversedirections will be zero exactly at the center (0.9 L) of thecavity (Fig. 1), and so in case of perpendicular injection atthis point there will no force along the horizontal direction.This will not affect the properties of the radiation spectrumand, regardless, such a situation is unlikely to be achieved inan experiment due its sensitivity to the initial conditions. Infact the nature of the magnetic mirror effect is to stabilizethe system leading to a robust experimental setup.

The trajectory of a 153 MeV (c0¼ 300) particle injectedat an angle hin¼p/2 at 0.5 L is presented in Fig. 3. In thiscase, we have chosen our parameters such that the centralmagnetic field is b0¼ 1200 and the magnetic field at the endsis bm¼ 1250. The time evolution of the z coordinate of theparticle is presented in Fig. 3(a), where it can be seen thatthe particle keeps bouncing between the two ends of thetube. The magnetic field at the ends (bm) influences the fre-quency of this oscillation. Figure 3(b) shows the temporalevolution of the parallel component of momentum (pk),which is directed along the z direction. The momentum staysconstant during the transit of the particle in the tube.However, when it reaches an end, the particle is reflectedback from the edge of the cavity and the momentum reverses

FIG. 2. The trajectories of the particlefor injection angles of p/6 (left col-umn) and p/3 (right column). Theorbits are too closely spaced to be indi-vidually resolved.



its sign. The temporal evolution of the perpendicular compo-nent of the momentum (p2

? ¼ p2x þ p2

y) is presented in Fig.3(c). Since there is only a slight difference in magnetic fieldstrength between the center and ends of the cavity, the varia-tion in p? is very small. Therefore, we have plotted the devi-ation from the arithmetic mean as a function of time. The p?increases slightly as the particle reaches the end, because themagnetic field is higher compared to the field in the center.The trajectory of the particle in the x-y plane is also shownin Fig. 3(d), where it can be seen that the orbits are very sta-ble in this plane, with a gyration radius given by rL¼ p?/b0+ 300/1200¼ 0.25. The magnetic mirror setup can beseen to be mimicking the dynamics of a particle in a verylong cavity with a uniform magnetic field.

IV. RESULTS AND DISCUSSION

In this section, we will study the effects of the electronenergy and the magnetic field strength on the properties of theemitted radiation. Before proceeding, we first stop to recon-sider the influence of the injection angle on the particle dy-namics. In Sec. III, we saw how perpendicular injection isessential if the particle is to maintain a very stable and uni-form orbit. It would seem likely that the more uniform theorbit the cleaner the corresponding radiation spectrum will be.In Fig. 4, we test this by plotting the spectra for the caseswhen a 153 MeV (c0¼ 300) electron is injected in a cavityhaving b0¼ 1200, bm¼ 4850 and 1250 for hin¼p/6 and p/2,respectively. It can be seen that, for the case of hin¼p/2 (Fig.4(b)), the spectrum is indeed very clean, peaking at exactlythe cyclotron frequency (b0/c0) corresponding to the centralmagnetic field. On the other hand, the spectrum for hin¼p/6(Fig. 4(a)) is a lot messier because there are a range of fre-quency components contributing due to the varying orbit ra-dius at the ends of the cavity. Since we are interested inutilizing this setup as a radiation source we will focus ourattention on perpendicular injection rather than oblique.

The oscillatory nature of the orbit makes it likely thatthere will be some similarities with the spectrum produced by

an electron in a plane wave field (see, for instance, Refs. 14and 15). However, despite similarities, the two cases are notequivalent. For example, the invariants B2 – E2, B % E of thetwo fields are not the same and it is never possible to boost toa frame where a plane wave becomes a constant B field.Nevertheless, we should keep in mind the various plane waveresults as we analyze the spectra. In particular, the high degreeof periodicity of the particle orbits will likely result in spectralpeaks of a very narrow bandwidth,16 and this is indeed con-sistent with what we have already seen in Fig. 4(b).

A. Spectrum by low energy electrons

We begin by considering the radiation emission spec-trum for (relatively) low energy electrons, such as those thatcould be generated using a conventional electron gun. Thespectrum as a function of the emission angle and the radiatedfrequency is plotted in Figures 5 and 6 for an incoming elec-tron of c0¼ 4, for two different magnetic field strengths. It

FIG. 3. The trajectory and momentumof an electron (c0¼ 300) injected at anangle hin¼p/2 into a cavity withb0¼ 1200, bm¼ 1250. The plots showthe temporal evolution of the z coordi-nate (a), the parallel component of themomentum pk (b), the perpendicularcomponent of the momentum p?(where pavg

? ¼ 299:991834) (c) and thetrajectory in the x-y plane (d).

FIG. 4. The radiation spectrum detected along the cavity for the case when153 MeV (c0¼ 300) electron is injected at p/6 (a) and at p/2 (b) to thecavity.



can be seen that in both cases the harmonics are very clean,appearing at integer multiples of the respective cyclotron fre-quencies (b0/c0). In the cases, we are considering there willbe many harmonics contributing to the spectrum, the totalnumber scaling like c3.1 Thus the total range of the spectrumcan, depending on the parameters, go from the THz to theXUV range. However, calculating the full spectrum is com-putationally expensive and so we have just plotted the firstfew harmonics here for the purposes of our discussion.Observe that the intensity of the radiation is roughly uniformover the whole angular range of each harmonic. This isbecause the angular widths of the emissions go like ,1/cwhich is of the same order of magnitude as the angular rangewe are considering.1 Note also that the fundamental (i.e.,first) harmonic is the only one that contributes preciselyalong the cavity axis hdetec¼ 0. This is consistent with theemissions in a plane wave (head-on collision), where thespectrum exhibits a “dead-cone” at hdetec¼ 0.15 As we moveto higher harmonics the direction they contribute in gradu-ally moves away from the cavity axis.

Since c0 is the same in these two cases, we can assessthe influence of the magnetic field strength on the propertiesof the spectrum. Because the harmonics are at integer multi-ples of the cyclotron frequency b0/c0, as we increase thestrength of the magnetic field the harmonics will be blueshifted. (At first sight, this appears to be the opposite to whathappens in the case of an electron in a plane wave field,where an increase in the wave amplitude causes the harmon-ics to be red shifted. However in this case, it is conventionalto define the field intensity such that it is normalized by thefrequency of the wave. In our case there is no fundamentalfrequency due to the background field—the timescales comefrom the size of the particle’s orbit. Since this is dependenton the c-factor, we are not comparing like with like.)

Next, we consider the dependence of the spectra on c0.Since we have already established that the emission harmon-ics occur at multiples of the cyclotron frequency we expect

that, for magnetic fields of a fixed strength, the frequenciesof the spectrum will be redshifted by a factor of 1/c0 as weincrease the electron energy. In Fig. 7, we investigate this byfixing the fields to be b0¼ 1200 and bm¼ 1250 and plottingthe fundamental harmonic in the hdetec¼ 0 direction forc0¼ 2, 3, 5, and 7. As expected, the frequencies of the har-monic peaks decrease as we increase the c-factor.

Finally, we will examine how the amplitude of the fun-damental harmonic changes with c0. It is known from thestandard textbook theory that a particle undergoing instanta-neous circular motion will radiate a power spectrum that hasthe form

dP , 1

c2

ðbBÞ2

1$ b sin hð Þ31$ cos2h

c2 1$ b sin hð Þ2

!; (10)

where we have abbreviated hdetec¼ h (see, for example,Chap. 14 of Ref. 1). Along the cavity axis the radiated power

FIG. 5. The emission spectrum for a particle of c0¼ 4 in a cavity withb0¼ 12, bm¼ 15. The center plot shows the intensity of the emitted radiationas a function of the detection angle hdetec and the dimensionless frequencyx/x0. The lower plot shows the radiated power as a function of frequency(summed over hdetec) and the right hand plot the radiated power as a functionof the angle (summed over frequency).

FIG. 6. The emission spectrum for a particle of c0¼ 4 in a cavity withb0¼ 1200, bm¼ 1250. The center plot shows the intensity of the emittedradiation as a function of the detection angle hdetec and the dimensionlessfrequency x/x0. The lower plot shows the radiated power as a function offrequency (summed over hdetec) and the right hand plot the radiated power asa function of the angle (summed over frequency).

FIG. 7. The radiation spectrum emitted by the gyrating particle withb0¼ 1200 and bm¼ 1250 for different energies. The fit line shows the varia-tion of the peak amplitude with particle energy. It is observed that the peakamplitude varies as 1/c2.



will go like (bB)2(1$ 1/c2)/c2 and so we can expect a 1/c2

scaling of peak emission amplitude with the particle energy.In Fig. 7, we plot a fit-line of the peak amplitude as a func-tion of particle’s energy. It is observed that amplitude indeeddecreases like 1/c2.

B. Spectrum by ultra-relativistic electrons

Next, we consider how the system behaves if we switchto an ultra-relativistic electron source, such as from a particleaccelerator. In Fig. 8, we show the emission spectrum for anelectron with c0¼ 600 inserted into the same cavity asbefore, b0¼ 1200 and bm¼ 1250. Once again the harmonicsare very clean, occurring at integer multiples of the cyclotronfrequency. The main difference in this case is that there is astrong peak of broadband radiation emitted in the directionperpendicular to the cavity (hdetec¼ p/2). This is because anultra-relativistic particle will radiate most of its energy in itsdirection of motion,1 which in this case will be tangential tothe circular orbit. We note that the amplitude of these har-monics is significantly lower than in the previous, lowerenergy cases. This is because the radiated power is moreconcentrated towards the perpendicular direction, as can beseen by considering Eq. (10) in the limit hdetec! p/2, b, 1.

Despite the peak radiation being perpendicular to thecavity, it can still be interesting to detect the radiation alongthe cavity axis, since here we have a clean frequency comb,rather than the “washed-out” broadband spectrum seen per-pendicular to the axis. However, the conversion efficiency isnot as high in this direction. In Fig. 9, we plot the fundamen-tal harmonic, viewed along the cavity axis (hdetec¼ 0), forvarious large c0’s. Once again, we see the same characteristicbehavior, with the peak positions being redshifted accordingto 1/c0 and the signal strength falling like 1=c2

0. As a result ofthis, the high c setup is not as practicable as a radiationsource as the lower energy (and more convenient) electron

gun scenario, though there will be higher harmonics extend-ing into the XUV range.

Nevertheless, the high energy setup may be interestingfrom a theoretical point of view, if we consider the effects ofRF on the particle dynamics. We will investigate this inSec. V.

C. Comparisons with gyrotrons

We have now studied the radiation emitted by both lowenergy and ultra-relativistic electrons in magnetic mirrorcavities and we have seen that various harmonics of the cy-clotron frequency can be generated using this setup, rangingfrom THz to XUV. A similar approach of radiation genera-tion is used in gyrotrons and so it is warranted to brieflycompare the two devices.

The gyrotron is a high power vacuum tube device whichis capable of generating radiation over the GHz to THz rangeby the principle of cyclotron resonance in strong magneticfields.17 The physics and dynamics of charged particles ingyrotrons are too involved to be addressed in detail in thismanuscript. Nevertheless, the helical motion of an electron ina magnetic field and the resulting radiation emitted by it formsthe underlying principle of both devices, thus warranting acomparison of the two, at least qualitatively. To make thecomparisons, we consider two descriptions of gyrotrons thatdescribed by Kumar et al.,18 where they consider the designand operation of a 35 GHz gyrotron, and that described byAshutosh et al.,19 where they consider the design and opera-tion of a 42 GHz gyrotron. A comparable set up for our mag-netic mirror cavity is to have a cavity length of 5 cm and anelectron energy of 65 keV (c0¼ 1.1272). For the comparisonwith the 35 GHz gyrotron, we use the same magnetic fieldstrength as specified in Ref. 18, 1.34 T (b0¼ 125), and for the40 GHz, 1.62 T (b0¼ 151).19 Figure 10 shows the radiationspectra for the two cases, detected along the cavity axis. It isobserved that the spectra peak at 33 and 40 GHz, respectively,for b0¼ 125 and b0¼ 151 (corresponding to the cyclotron fre-quency). This is in good agreement with the results presentedin Refs. 18 and 19, though we are off by 2 GHz in both cases

FIG. 8. The emission spectrum for a particle of c0¼ 600 in a cavity withb0¼ 1200, bm¼ 1250. The center plot shows the intensity of the emittedradiation as a function of the detection angle hdetec and the dimensionlessfrequency x/x0. The lower plot shows the radiated power as a function offrequency (summed over hdetec) and the right hand plot the radiated power asa function of the angle (summed over frequency).

FIG. 9. The radiation spectrum emitted by the gyrating particle withb0¼ 1200 and bm¼ 1250 for different energies. The fit line shows the varia-tion of the peak amplitude with particle energy. It is observed that the peakamplitude varies as 1/c2.



due to the fact that the particle dynamics will be slightly dif-ferent in our cavity compared to those in a gyrotron. (It shouldbe noted that in the rest of this article our conventions use theangular frequency x¼ 2pc/k but, for the purposes of thesecomparisons, we have plotted Figure 10 in terms of the ordi-nary frequency !¼x/2p.)

Finally, we comment on the relative efficiency of thetwo devices. In the case of the gyrotron, the length of thecavity plays an essential role because the electron only tra-verses this distance once before it leaves the system. In themagnetic mirror cavity, because of the mirroring effect, theeffective length of the cavity depends on the simulationtime. This clearly makes our set up more efficient since theelectrons are recirculated; however, it does not directlyequate to a higher emission amplitude since there will bepauses in the emissions, and the spectrum will be split intotwo distinct peaks caused by the Doppler shift, when themotion of the guiding center changes direction each time theelectron is reflected off a cavity end.

It is also worth making some comparisons with freeelectron lasers (FELs), since these also operate by passing anelectron through a magnetic field. However, the field struc-ture a FEL is more complicated, being comprised of a peri-odic arrangement of magnets with alternating poles (anundulator). The relationship between the wavelength of theradiation emitted by the electron and the spatial period of themagnetic field (undulator wavelength), ku, is20

k ¼ ku

2c21þ eBkuffiffiffi

8p

pmc

! "2 !

: (11)

As an illustrative example for typical parameters we havebeen considering (e.g., b0¼ 1200, c¼ 5 resulting in a fre-quency of 72 GHz), one would require an undulatory spacingof about 3.1 mm in order to obtain the same frequency signal.

V. EFFECT OF RADIATION FRICTION

Since the electron in the cavity is radiating, it must alsobe losing energy. These energy loses will alter the motion of

the particle; an effect known as radiation reaction or radia-tion friction (RF). Over short timescales the amount of radia-tion loss is typically small compared to the particle’s energyand so can ordinarily be neglected. However, if we considerthe trajectory of the particle over an extended time periodthen the (instantaneously) small effect of radiation loss willhave more of a cumulative impact. Additionally, if weincrease the strength of the magnetic field, or increase the c-factor, then the acceleration acting on the particle may bestrong enough for RF effects to become important overshorter timescales. It is worth extending our modeling toinclude RF for two reasons. First, so that we can checkwhether it is valid to neglect RF in the regimes we have beenconsidering and, second, to explore the potential for usingthis setup with different parameters to investigate RF in itsown right.

In classical electrodynamics, the effect of RF on themotion of a particle can be included by means of a correc-tional term to the Lorentz force. This correction termdescribes the energy loss to the particle when it is acceler-ated. However, despite having been studied for over 100years, finding the correct equation to describe the dynamicsof a radiating particle remains one of the most fundamentalproblems in electrodynamics. The most common startingpoint is to form a coupled system comprised of the Lorentzand Maxwell’s equations for the radiating particle. The solu-tion of this is known as the Lorentz-Abraham-Dirac equa-tion21–23 and gives a correction to the Lorentz force which isa third order time derivative of the particle position. Becauseit is third order the equation suffers notorious defects such aspre-acceleration and (unphysical) runaway solutions. One ofthe most common ways of removing these problems is toadopt a perturbative approximation such as that formulatedby Landau and Lifshitz24 so that the equation of motionbecomes

dp

dt¼ fL þ fRF; (12)

where fL is the Lorentz force (4) and the radiative correctionterm, expressed in terms of the background magnetic field, is

fRF ¼ $4

3p

re

k0

! "c v& @

@tþ v %r

! "B

& '

þ 4

3p

re

k0

! "v& B& B½ ( $ 4

3p

re

k0

! "c2 v& Bð Þ2h i

v;

(13)

where re) e2/mc2+ 2.8& 10$15m is the classical electron ra-dius. Equation (13) is valid as an approximation to LADwhen the radiative reaction force is much less than theLorentz force in the instantaneous rest frame of the particle.We note that there are numerous alternative equations in theliterature (see, for example, Refs. 25 and 26) and it is still anopen problem as to which is the correct formulation.However, the Landau Lifshitz equation has, along with someothers, recently been shown to be consistent with quantumelectrodynamics to the order of the fine structure constanta.27,28 Despite these controversies, no experiment has been

FIG. 10. The radiation spectrum from a 65 keV electron in magnetic fieldsof strength b0¼ 125 and b0¼ 151. These cases are similar to the gyrotronsreported in Refs. 18 and 19.



conducted to distinguish between the different models. Somestudies have investigated the use of high intensity lasers toprobe RF effects (see, for example, Refs. 29 and 30), butsuch an approach may not be the most optimum for distin-guishing between models, because particles inserted into thebeam will radiate away much of their energy before theyenter the most intense part of the field.31,32 This is why anexperiment using our magnetic mirror cavity may be moreadvantageous, since the particle will remain trapped in thefield for an extended time period.

To implement the RF effects into our simulation, weupdate our particle pusher to include the terms (13) in theequation of motion. The first (derivative) term is smallerthan the other two since it is linear in the magnetic fieldstrength while the others are quadratic. Thus, in order to savecomputation time, we neglect this term in our simulations.

In Fig. 11, we show the spectrum (and, in the inset, thec-factor) for the case when c0¼ 400 with the RF termsincluded. It can be seen from the inset that the particle gradu-ally looses energy as it moves back and forth along the cav-ity, falling to about two thirds of its initial value by the endof the simulation (s¼ 1& 105 is equivalent to 0.053 ms).From the spectrum, we can see that initially the emissionpeak is at the same value as the case without RF (i.e.,x/x0¼ 3), just as we would expect. Then, as the particleloses energy, the frequency of the spectral peak graduallyincreases, resulting in a broadening of the harmonic as com-pared to the case without RF. This behavior is consistentwith Fig. 9 where we found that particles with a lower c-fac-tor radiate at higher frequencies along the cavity axis.

Thus, we can see that, when our high-c magnetic cavitysource is used over an extended time period, the frequencyharmonics of the emitted radiation are slowly blue shifted asthe electron loses energy due to RF. Although the broadeningof the harmonic due to radiation loss is not large, it is never-theless significant and easily measurable. This raises theprospect of comparing the spectra calculated using the vari-ous competing classical RF theories with the experimentalresults, potentially allowing us to test the different models.(This is not completely dissimilar from the idea of cyclingan electron through a modified Penning trap, which was pro-posed in Ref. 33.) However, such an investigation is beyond

the scope of this paper and so we leave it for a futurepublication.

VI. CONCLUDING REMARKS

We have presented (to best of our knowledge) a previouslyunconsidered setup for generating the radiation in THz range.The setup utilizes a magnetic mirror to confine a (relativistic)charged particle to a cavity. The particle is injected perpendicu-larly to the applied magnetic field and then the “magnetic-mirror” effect serves to mimic the dynamics of particle in avery long cavity with uniform magnetic field. The result is auniform, closely spaced, circular orbit (helical motion) for theparticle, causing it to emit a narrow band of radiation at its cy-clotron frequency. The emitted radiation will be in the THzrange and can be tuned to an exact frequency but changing theparticle energy. The simplicity and portability of the proposedsetup makes it a viable THz radiation source which would beuseful for medical research, defense applications, communica-tion purposes, imaging, and spectroscopy. Furthermore, at non-zero detection angles higher harmonics of the fundamentalfrequency (cyclotron frequency) are also observed. This prom-ises to yield a frequency comb from the THz to the XUV range(although we have only presented the first few harmonics dueto limitations in computational resources).

For convenience, a comparison of the dimensionlessunits used in this article to that of SI units is presented inTable I. It can be seen increasing the energy of the particlereduces the cyclotron frequency (!c¼ c b0/c). Also, a higherparticle energy leads to a larger larmor radius (rL), which isan essential parameter for design considerations.

The effect of RF on the emitted radiation has also beenstudied. It is observed that the spectrum becomes blueshiftedwhen RF is taken into account. This is because, as the parti-cle looses energy, its cyclotron frequency is reduced by 1/c(where here c would be a function of time). Thus, apart fromuse as a THz radiation source, the setup could also beexploited to experimentally validate the various competingtheories of RF. The extensive analysis of the THz radiationand its harmonics under the different models is beyond thescope of the current manuscript and has been left for a futurepublication.

ACKNOWLEDGMENTS

A.H. acknowledges the Science and EngineeringResearch Board, Department of Science and Technology,Government of India for funding the Project SR/FTP/PS-189/2012. C.H. acknowledges support from the EPSRC, UK,Grant No. EP/I029206/1-YOTTA.

FIG. 11. Spectrum for the case c0¼ 400, hin¼p/2, hdetec¼ 0, b0¼ 1200, andbm¼ 1250. Here, RF is included. The fit line again shows a 1/c2 scaling.

TABLE I. Comparing dimensionless units to SI units.

c E ðMeVÞ b0 B (T) rL (cm) !c (,GHz) 10!c (,THz)

1.5 0.255 500 5.35 0.0477 100.0 1.0

3.0 1.022 500 5.35 0.0955 50.0 0.5

5.0 2.044 500 5.35 0.1591 30.0 0.3

5.0 2.044 1200 12.85 0.0663 72.0 0.72

7.0 3.066 1200 12.85 0.0928 51.4 0.514



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Magnetic mirror cavities as terahertz radiation sources and a means of quantifying radiation...

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