Theory of electromagnetic insertion devices and the correspondingsynchrotron radiation

Muhammad Shumail* and Sami G. Tantawi†

SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA(Received 15 April 2016; published 27 July 2016)

Permanent magnet insertion devices (IDs), which are the main radiation generating devices insynchrotron light sources and free-electron lasers, use a time-invariant but space-periodic magneticfield to wiggle relativistic electrons for short-wavelength radiation generation. Recently, a high powermicrowave based undulator has also been successfully demonstrated at SLAC which promises theadvantage of dynamic tunability of radiation spectrum and polarization. Such IDs employ transverseelecromagnetic fields which are periodic in both space and time to undulate the electrons. In this paper wedevelop a detailed theory of the principle of electromagnetic IDs from first principles for both linear andcircular polarization modes. The electromagnetic equivalent definitions of undulator period (λu) andundulator deflection parameter (K) are derived. In the inertial frame where the average momentum of theelectron is zero, we obtain the figure-8-like trajectory for the linear polarization mode and the circulartrajectory for the circular polarization mode. The corresponding radiation spectra and the intensity ofharmonics is also calculated.

DOI: 10.1103/PhysRevAccelBeams.19.074001

I. INTRODUCTION

The conventional insertion devices (IDs) (wigglers orundulators) consist of a series of alternating magnetic polesthat induce a periodic local deflection in a relativisticelectron beam of a synchrotron to generate intense radiationin a specific range of the spectrum. The IDs are charac-terized as either wigglers or undulators with no funda-mental difference between the two. In undulators, thedeflection is weak and the radiation interference is morepronounced as compared to the wigglers [1]. The undu-lators were first proposed in 1951 [2] and then demon-strated in 1953 [3] by Motz. The undulators can also beused after a linear accelerator (linac), to generate a coherentfree-electron laser radiation, as first demonstratedby Madey in 1977. Today, the technology of permanentmagnet undulators (PMUs) has become an established artand is being pushed to its limits to satisfy the requirementsof the fourth generation light sources. The prime examplesof such fourth generation light sources are the LinacCoherent Light Source (LCLS) [4] and the SPring-8Angstrom Compact Free-Electron Laser (SACLA) [5].Despite the great success of these modern x-ray sources,the intrinsic nature of the magnetostatic fields poses many

limitations. Permanent magnets do not allow the muchsought after feature of shorter undulator period, whichallows for the economical lower energy systems, whilemaintaining adequate aperture for the electron beam andwithout compromising on the field strength [6]. Also, thedynamic control is slow and limited with PMUs. Fastdynamic control of the radiation can offer exciting scien-tific opportunities. These limitations could be overcome bythe use of high-power guided microwaves to produce aperiodical transverse wiggling field. Stimulated emissionby a microwave cavity from a relativistic beam wasoriginally suggested in 1968 [7]. But due to the challengesassociated with the confinement and control of high-powermicrowaves, the implementation of a state-of-the-art micro-wave based insertion device has remained elusive for a longtime. Recently, however, we have demonstrated a practical,tunable microwave-based undulator with undulator periodλu ¼ 13.9 mm and undulator deflection parameter K ¼ 0.7[8]. With the revival of this interest in the implementationof tunable electromagnetic IDs, we are presenting athorough investigation of the specific principle of suchdevices to serve as a theoretical reference and benchmarkfor similar devices.The conventional magnetic undulators rely on the

transverse magnetic fields, alternating in space but staticin time, to undulate or wiggle the charged particles, usuallyelectrons, which are generally moving at relativistic speeds.If the primary direction of motion of the electrons is takento be along the z-axis then a magnetic field directed alongthe y-axis but alternating as a function of the z-axis wouldcause the electrons to wiggle along the x-axis. This leads toa narrow beam of radiation along the forward direction of

*Present address: Habib University, Karachi.shumail@alumni.stanford.edu;

†tantawi@slac.stanford.edu

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

PHYSICAL REVIEW ACCELERATORS AND BEAMS 19, 074001 (2016)

2469-9888=16=19(7)=074001(12) 074001-1 Published by the American Physical Society

the electrons [9]. Depending upon the relativistic speed ofthe electrons, the wavelength of this radiation is orders ofmagnitude smaller than the period of the magnetic fieldalternation along the z-axis [9,10].The charged particles can also be wiggled by time

harmonic electromagnetic waves. In this case, both theelectric and magnetic fields would contribute to wiggle theelectrons. Another feature of this scheme would be thatthe fields not only alternate in space but time as well. Adevice employing electromagnetic waves to undulate theelectrons would be called an electromagnetic undulator.The electromagnetic fields in the frequency range of theradio frequency (rf) or microwave bands of the spectrumare more likely to be used for this purpose as the technologyof generating and manipulating the electromagnetic fieldsis more mature in this range of the spectrum. Hence, insteadof electromagnetic undulators, we generally use the term rfor microwave undulators.The principle of microwave undulators to produce

synchrotron radiation can be best understood in the inertialframe where average electron velocity is zero. In this frame,the electron causes Thomson scattering of the Doppler-shifted microwaves. The scattered radiation, Doppler-shifted back to the lab frame is the highly directionaland very high frequency synchrotron radiation.In this paper, we explain the fundamental principles of a

microwave undulator and derive the equivalent mathemati-cal definitions of various parameters, like λu and K, thatare also used to characterize the conventional magneticundulators. Earlier, Batchelor has derived the equivalentequations of λu and K for linear polarized TE modes inrectangular waveguides [11]. Pellegrini has also done asimilar analysis for the particular case of circular polarizedfundamental TE mode in a square waveguide [12].However, our analysis is more generic and we take theseequations further to obtain the trajectory of an electron forboth linear and circular polarization. Moreover, we havealso investigated the spectrum of the synchrotron radiationand the far-field intensity of harmonics versus the obser-vation angle. For both linear and circular polarizationbalanced hybrid HE1n modes, the electron trajectory in amicrowave undulator and hence the corresponding radia-tion characteristics turn out to be similar to those of PMUs.

II. ELECTROMAGNETIC FIELDS ON AXIS

It has been discussed that the dipole modes, like TM1n,TE1n, and HE1n, are the most simple and interesting modesfor microwave undulator application [13]. For these modesthe fields, near the vicinity of the axis (z-axis) of the devicealong which the electrons are supposed to travel, aretransverse and their expressions are given in the followingequations, where ω is the angular rf frequency, β is thepropagation constant, t is time, and Eaxis and Baxis are theelectric and magnetic fields, respectively, on or nearthe axis.

Linear polarization.—

Eaxisðz; tÞ ¼ −E⊥ sinðβzÞ cosðωtÞx

¼ E⊥sinðωt − βzÞ − sinðωtþ βzÞ

2x ð1aÞ

Baxisðz; tÞ ¼ B⊥ cosðβzÞ sinðωtÞy

¼ B⊥sinðωt − βzÞ þ sinðωtþ βzÞ

2y: ð1bÞ

Circular polarization.—

Eaxisðz; tÞ ¼ −E⊥ sinðβzÞ½cosðωtÞxþ sinðωtÞy�

¼ E⊥sinðωt − βzÞ − sinðωtþ βzÞ

2x

− E⊥cosðωt − βzÞ − cosðωtþ βzÞ

2y ð2aÞ

Baxisðz; tÞ ¼ B⊥ cosðβzÞ½cosðωtÞxþ sinðωtÞy�

¼ B⊥sinðωt − βzÞ þ sinðωtþ βzÞ

2y

þ B⊥cosðωt − βzÞ þ cosðωtþ βzÞ

2x: ð2bÞ

It is evident that E⊥ and B⊥ are the peak standing waveamplitudes of the respective fields. We will also use thesymbols for the free space rf wavelength (λ) and the wavenumber (k≡ 2π=λ).We are interested to explore the motion of an electron

under the influence of these fields which have beendescribed in the lab inertial frame. The electron moves ata relativistic speed along the positive z-axis. It turns outthat there is another inertial frame, defined as electronframe, in which its motion is a stable closed orbit. Ourstrategy is to first solve the equations of motion for theelectron in the electron frame. We choose the axes of theelectron frame to coincide with those of the lab frame attime t ¼ 0. Moreover, the electron frame is assumed tobe moving with a velocity v0z with respect to the labframe. Thus, as per our definition of the electron framewhere the electron moves in a closed orbit, v0z is alsothe average velocity of electron in the lab frame. Sincethe electron is moving at a relativistic speed, therefore v0is very close to c, the speed of light in vacuum. Torepresent a quantity in the electron frame, we will add aprime ( 0) to the corresponding symbol of that quantity inthe lab frame.Lorentz transformations [14] yield the following expres-

sions for the fields in the electron frame:

z ¼ γ0ðz0 þ v0t0Þ ð3aÞ

t ¼ γ0ðt0 þ z0v0=c2Þ ð3bÞ

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E0axisðz0; t0Þ ¼ γ0fEaxisðz; tÞ þ cz × Baxisðz; tÞg ð3cÞ

B0axisðz0; t0Þ ¼ γ0

�Baxisðz; tÞ −

zc× Eaxisðz; tÞ

�; ð3dÞ

where γ0 ≡ 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − v20=c

2p

.

III. LINEAR POLARIZATION

Using v0 ≈ c and Eq. (1), Eq. (3) can be written as

E0axisðz0; t0Þ ¼ −cγ0Bu½sinfω0

radðt0 þ z0=cÞgþ δ sinfϵω0

radðt0 þ z0=cÞg�x ð4aÞ

B0axisðz0; t0Þ ¼ γ0Bu½sinfω0

radðt0 þ z0=cÞgþ δ sinfϵω0

radðt0 þ z0=cÞg�y; ð4bÞ

where

Bu ≡ B⊥ þ E⊥=c2

ð5aÞ

ω0rad ≡ 2πcγ0

λuð5bÞ

λu ≡ λ

1þ β=kð5cÞ

ϵ≡ 1 − β=k1þ β=k

ð5dÞ

δ≡ B⊥ − E⊥=cB⊥ þ E⊥=c

¼8<:

ϵ TM1n

−ϵ TE1n1−Λ1þΛ ϵ HE1n

ð5eÞ

and Λ is the mode hybrid factor.In Eq. (4), the terms proportional to δ are due to the

forward traveling wave, while the other ones are due to thebackward traveling wave. When only backward travelingwave is present or when δ ≪ 1, which is usually the case,the orbital angular frequency of the electron, in the electronframe, would be ω0

rad as this is the angular frequency of thedriving electromagnetic field. Thus, the electron yields aradiation with angular frequency ω0

rad in the electron frame.To obtain the angular frequency of this radiation in the labframe, as observed at an angle θ ≪ 1 from the axis, weemploy Eq. (A3):

ωrad ¼ ω0rad

2γ01þ ðγ0θÞ2

: ð6Þ

The corresponding wavelength of the radiation in the labframe is given as

λrad ≡ 2πcωrad

¼ λu2γ20

½1þ ðγ0θÞ2�: ð7Þ

Using −e and m for the charge and mass of the electron,respectively, the equations of motion for the electron in theelectron frame can be written in terms of the rate of changeof momentum (p0) and energy (T 0) as [15]

dp0

dt0¼ −eðE0

axis þ v0 × B0axisÞ ð8aÞ

dT 0

dt0¼ −ev0 · E0

axis: ð8bÞ

Here, v0 is the velocity of the electron along the closedorbit in the electron frame. Since there is no force along they-axis, the motion of the electron will remain confined inthe z0 − x0 plane if it does not have any initial velocitycomponent along the y-axis. Also note that though Eq. (8b)is not an independent equation and can be obtained fromEq. (8a), it will help us simplify the analysis.Now we substitute Eq. (4) in Eq. (8):

dγ0mv0xdt0

¼ eγ0c

�1þ v0z

c

�Bu

�sin

�ω0rad

�t0 þ z0

c

��

þ δ sin

�ϵω0

rad

�t0 þ z0

c

���ð9aÞ

dγ0mv0zdt0

¼ −eγ0v0xBu

�sin

�ω0rad

�t0 þ z0

c

��

þ δ sin

�ϵω0

rad

�t0 þ z0

c

���ð9bÞ

dγ0mc2

dt0¼ eγ0cv0xBu

�sin

�ω0rad

�t0 þ z0

c

��

þ δ sin

�ϵω0

rad

�t0 þ z0

c

���; ð9cÞ

where

γ0 ≡ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − v02x þv02z

c2

q : ð10Þ

Multiplying Eq. (9b) by c and adding to Eq. (9c) yields aconstant of motion:

A ¼ γ0�1þ v0z

c

�> 1: ð11Þ

Corresponding to the velocity, v, of the electron inthe lab frame we define γ ≡ 1ffiffiffiffiffiffiffi

1−v2c2

p . Note that γ is the energy

of the electron measured in units of electron mass

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(mc2 ≈ 0.511 MeV). According to the Lorentz transforma-tions, the energy in the lab frame is given in terms of theenergy and momentum in the electron frame as follows:

γmc2 ¼ γ0ðγ0mc2 þ v0γ0mv0zÞ ¼ γ0γ0mc2

�1þ v0

cvzc

�:

ð12Þ

Using v0 ≈ c and Eq. (11), we get

γ0 ¼ γ=A: ð13Þ

Substituting Eq. (13) in Eq. (7) yields

λrad ¼λu2γ2

½A2 þ ðγθÞ2�: ð14Þ

Before solving the equations of motion, let us definesome unitless quantities for the sake of notational ease:

u0x ≡ v0x=c ð15aÞ

u0z ≡ v0z=c ð15bÞ

τ0 ≡ ω0radt

0 ¼ 2πcγAλu

t0 ð15cÞ

ξ0 ≡ ω0rad

cx0 ¼ 2πγ

Aλux0 ¼ ξ0ð0Þ þ

Zτ0

0

u0xðτ00Þdτ00 ð15dÞ

η0 ≡ ω0rad

cz0 ¼ 2πγ

Aλuz0 ¼ η0ð0Þ þ

Zτ0

0

u0zðτ00Þdτ00 ð15eÞ

K ≡ e2πmc

λuBu ¼ 0.934λuðcmÞBuðTÞ: ð15fÞ

The normalized spatial coordinates (ξ0 and η0) can beobtained by integrating the corresponding normalizedvelocities (u0x and u0z) with respect to the normalized timevariable τ0. The dimensionless parameterK is the undulatordeflection parameter defined in terms of the equivalentundulator period λu and the equivalent magnetic field Bu.Now, using Eq. (15) we obtain the normalized versionof Eqs. (9a), (9b), and (11) as follows:

dγ0u0xdτ0

¼Kð1þu0zÞ½sinðτ0 þη0Þþδsinfϵðτ0 þη0Þg� ð16aÞ

dγ0u0zdτ0

¼ −Ku0x½sinðτ0 þ η0Þ þ δ sinfϵðτ0 þ η0Þg� ð16bÞ

A ¼ γ0ð1þ u0zÞ > 1: ð16cÞ

Equation (16a) can exactly be integrated to yield

γ0u0x ¼ −K�cosðτ0 þ η0Þ þ δ

ϵcosfϵðτ0 þ η0Þg

�: ð17Þ

The integration constant was chosen to be zero, as bydefinition the average momentum vanishes in the electronframe. We can use Eqs. (10) and (16c) to express thenormalized velocities (u0z and u0x) in terms of the normalizedenergy (γ0):

u0z ¼Aγ0− 1 ð18aÞ

u0x ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Aγ0 − ðA2 þ 1Þ

pγ0

: ð18bÞ

Equations (17) and (18b) imply

γ0 ¼ A2 þ 1þ K2

2½1þðδϵÞ2�

2A

þK2

4A

�cosf2ðτ0 þ η0Þgþ 4

δ

ϵcosðτ0 þ η0Þcosfϵðτ0 þ η0Þg

þ�δ

ϵ

�2

cosf2ϵðτ0 þ η0Þg�: ð19Þ

Now, consider the longitudinal momentum that can beobtained from Eqs. (18a) and (19):

γ0u0z¼A2− ½1þK2

2f1þðδϵÞ2g�

2A

−K2

4A

�cosf2ðτ0 þη0Þgþ4

δ

ϵcosðτ0 þη0Þcosfϵðτ0 þη0Þg

þ�δ

ϵ

�2

cosf2ϵðτ0 þη0Þg�: ð20Þ

Since the average longitudinal momentum should bezero in the electron frame, we get the following usefulresult from Eq. (20):

A2 ¼ 1þ K2

2

�1þ

�δ

ϵ

�2�: ð21Þ

For the balanced hybrid HE1n modes, Λ ≈ 1 ⇒ δϵ ¼

1−Λ1þΛ ≈ 0 ⇒ A2 ¼ 1þ K2

2. Hence, Eq. (14) becomes

λrad ¼λu2γ2

�1þ K2

2þ ðγθÞ2

�: ð22Þ

Note that Eqs. (15f) and (22) are analogous to thecorresponding equations for PMUs [16]. Hence, we callBu and λu in Eq. (15f) the equivalent magnetic field and theequivalent undulator period, respectively.

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Now, with the assumption of the balanced hybrid HE1n

modes (δϵ ¼ 0), we get the following equations of motionwhich are the same as that for the PMUs:

γ0 ¼ 4þ 2K2 þ K2 cosf2ðτ0 þ η0Þg4

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ K2

2

q ð23aÞ

u0x ¼4K

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ K2

2

qcosðτ0 þ η0Þ

4þ 2K2 þ K2 cosf2ðτ0 þ η0Þg ð23bÞ

u0z ¼−K2 cosf2ðτ0 þ η0Þg

4þ 2K2 þ K2 cosf2ðτ0 þ η0Þg ð23cÞ

ξ0 ¼Z

τ0

0

u0xðτ″Þdτ″ ð23dÞ

η0 ¼Z

τ0

0

u0zðτ″Þdτ″: ð23eÞ

Here, we have chosen ξ0ð0Þ ¼ η0ð0Þ ¼ 0. It is evidentfrom Eq. (23) that the motion parameters in the electronframe are a periodic function of τ0, where u0zðτ0Þ, η0ðτ0Þ, andγ0ðτ0Þ have a period of π while u0xðτ0Þ and ξ0ðτ0Þ exhibit aperiod of 2π. It turns out that for K ≤ 1, it is a goodapproximation to replace η0, when it appears in the argu-ment of cosine function in Eq. (23), by its fundamentalharmonic only:

η0 ≈D2 sinð2τ0Þ; ð24Þ

where, using an iterative procedure, we get the followingempirical equation for the constant D2:

D2 ¼ −0.1385K2 þ 0.0546K3: ð25Þ

Thus for K ≤ 1, we need to solve the following integrals,which can easily be performed numerically, to get thenormalized trajectory of the electron in the electron frame:

ξ0ðτ0Þ ¼Z

τ0

0

4Kffiffiffiffiffiffiffiffiffiffiffiffiffi1þ K2

2

qcos½τ″ þD2 sinð2τ″Þ�

4þ 2K2 þ K2 cos½2fτ″ þD2 sinð2τ″Þg�dτ″

ð26aÞ

η0ðτ0Þ ¼Z

τ0

0

−K2 cos½2fτ″ þD2 sinð2τ″Þg�4þ 2K2 þ K2 cos½2fτ″ þD2 sinð2τ″Þg�

dτ″:

ð26bÞ

It turns out that for K ≤ 1, we can replace the moreprecise Eqs. (23b), (23c), and (26) with the followingapproximate equations which are more intuitive:

u0xðτ0Þ ≈Kffiffiffiffiffiffiffiffiffiffiffiffiffi1þ K2

2

q cosðτ0Þ ð27aÞ

u0zðτ0Þ ≈ −K2

2ð2þ K2Þ cosð2τ0Þ ð27bÞ

ξ0ðτ0Þ ≈ Kffiffiffiffiffiffiffiffiffiffiffiffiffi1þ K2

2

q sinðτ0Þ ð27cÞ

η0ðτ0Þ ≈ −K2

4ð2þ K2Þ sinð2τ0Þ: ð27dÞ

Equations (27c) and (27d) are the parametric equationsof the figure-8-like Lissajous curve. In the literature relatedto PMUs, we usually find mention of Eqs. (27a) and (27b),which are approximate, rather than the more precise versiongiven by Eqs. (23b) and (23c) (see, for example, [17]). Itturns out that, though qualitatively both sets of equationsyield similar radiation characteristics, there are hugequantitative differences for higher harmonics. In this paper,we have used the more precise version of equations todetermine angular and spectral radiation characteristics.Figure 1 shows the orbits of the electron in the electronframe for K ¼ 0.5 (blue) and K ¼ 1.0 (red, dashed). Notethat the horizontal coordinate is stretched 10 times incomparison to the vertical coordinate. Though these orbitswere obtained using the more precise Eq. (26), we getvisually overlapping curves if we use the approximate butintuitive version given by Eqs. (27c) and (27d).Now we will employ standard Lorentz transformations

to find the corresponding motion parameters in the lab

FIG. 1. Normalized orbit of the electron in the electron inertialframe for two different values of K. At τ0 ¼ 0; 2π; 4π;… theelectron crosses the origin to enter the second quadrant from thefourth. At τ0 ¼ π; 3π; 5π;… the electron crosses the origin toenter the third quadrant from the first.

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frame. Here we will use the result derived in Eq. (13), i.e.,

γ0 ¼ γ=A and γ0v0=c ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðγAÞ2 − 1

q:

ξ≡ 2π

λux ¼ A

γξ0 ð28aÞ

ζ ≡ 2π

λuy ¼ 0 ð28bÞ

η≡ 2π

λuz ¼ η0 þ τ0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

�Aγ

�2

sð28cÞ

τ≡ 2πcλu

t ¼ τ0 þ η0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

�Aγ

�2

sð28dÞ

ux ≡ vxc¼ Au0x

γh1þ u0z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðAγÞ2

q i ð28eÞ

uz ≡ vzc¼

u0z þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðAγÞ2

q1þ u0z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðAγÞ2

q ð28fÞ

αx ≡ λu2πc2

dvxdt

¼ duxdτ

¼ 1

1þ u0zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðAγÞ2

q duxdτ0

ð28gÞ

αz ≡ λu2πc2

dvzdt

¼ duzdτ

¼ 1

1þ u0zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðAγÞ2

q duzdτ0

: ð28hÞ

We can use Eqs. (28a) and (28c) to calculate thetrajectory of the electron in the lab frame. However, usingEqs. (28a) and (27c), along with the fact that for relativisticmotion η ≈ τ ≈ τ0, we obtain following simple equation, so-called second field integral, for the trajectory of the electronin the lab frame:

ξ ¼ KγsinðηÞ: ð29Þ

As expected, Eq. (29) is the same as for the case ofPMUs [18]. To calculate the energy of the electron as afunction of time in the lab frame, we can use Eqs. (28e)and (28f). Figure 2 shows the electron trajectory and theslight modulation in the energy of the electron in the labframe for K ¼ 0.5 (blue) and K ¼ 1.0 (red, dashed). Theperiod chosen corresponds to the interval 0 ≤ τ0 ≤ 2π. ForFig. 2(b) the value of γ was chosen corresponding to anelectron energy of 60 MeV (γ ¼ 117.417).Having solved the trajectory, we can calculate the

electric field (Erad) of the far-field radiation at a space-time point ðx0; y0; z0; ct0Þ≡ λu

2π ðξ0; ζ0; η0; τ0Þ in the labframe using the expression derived from Liénard-Wiechert potentials [19]:

Eradðξ0; ζ0; η0; τ0Þ ¼ −πeϵ0λ

2u

r × fðr − urÞ × αgðr − u:rÞ3 ð30aÞ

τ0 ¼ τ þ r; ð30bÞ

where

r≡ ðξ0 − ξÞxþ ðζ0 − ζÞyþ ðη0 − ηÞz ð31aÞ

r≡ jrj ð31bÞ

u≡ uxxþ uyyþ uzz ð31cÞ

α≡ αxxþ αyyþ αzz: ð31dÞ

Here, ϵ0 is the free space permittivity. Note that for thecurrent case (linear polarization) ζ ¼ uy ¼ αy ¼ 0.These equations can be used to calculate, for example,

the angular and spectral distribution of intensity of variousharmonics at any far-field point due to a single electron.

(a)

(b)

FIG. 2. Trajectory and energy modulation in the lab frame.(a) A period of electron trajectory. (b) Normalized change in theenergy of the electron during a single period.

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Figure 3 shows the intensity of various harmonics (Hn)versus the off-axis angle θ in a far-field transverse plane,100λu ahead of the electron. The electron energy is chosento be 60 MeV (typical energy of the electron beam atNLCTA, SLAC). The subscript onH represents the numberof harmonic. It is evident that while odd harmonics aremore intense on the axis, the even harmonics are moreintense off axis. Moreover, the intensity of any evenharmonic along the rf E plane is an order of magnitudehigher than its intensity along the rf B plane. Figure 4shows the on-axis intensity of various odd harmonicsagainst different values of K. Generally, the intensity ofthe harmonics increases as we increase K. At higher valuesof K, the intensity of higher harmonics increases at theexpense of the intensity of the lower harmonics. Note thatthe intensity of the odd harmonics H3 and H5 increases by

an order of magnitude as we go from K ¼ 0.5 to K ¼ 1.0.However, the intensity of the fundamental harmonic ismaximum at K ≈ 0.75.We can also calculate the spectrum of the radiation as

observed along a line in the far-field transverse plane bycalculating the intensity and the wavelength of the funda-mental harmonic along that line. According to Eq. (22) thewavelength increases as we go off axis. Figure 5 shows thespectra calculated for K ¼ 0.5 (blue) and K ¼ 1.0 (red,dashed). For these calculations, γ ¼ 117.417 (correspond-ing to 60 MeV electron energy) and λu ¼ 1.39 cm with 78undulator periods as per our demonstrated microwaveundulator [8]. The spectrum was calculated assuming theobservation line to be along x ¼ y in the far-field transverseplane. It has been observed, however, that the orientation ofthe observation line does not much change the profile of the

(a) (b)

(c) (d)

(e) (f)

(g) (h)

FIG. 3. Linear polarization: For a single electron the intensity (in units of 10−12 Wm2=λ4u) at a distance of 100λu of the first three oddand even harmonics vs the off-axis angle θ in a far-field transverse plane. The electron energy is 60 MeV.

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spectrum. Each spectrum shown in Fig. 5 is in fact theconvolution of the so-called line function spectrum, cor-responding to the duration of 78 undulating cycles, with thecorresponding continuous theoretical spectrum. The on-axis radiation wavelength as determined by Eq. (22) is567 nm (756 nm) corresponding to K ¼ 0.5 (K ¼ 1.0).Note that the peaks of the corresponding spectra occur alittle right to these wavelengths which are marked bydashed vertical lines in Fig. 5.

IV. CIRCULAR POLARIZATION

Using v0 ≈ c and Eq. (2), Eq. (3) can be written as

E0axisðz0; t0Þ ¼ −cγ0Bu

��sinfω0

radðt0 þ z0=cÞgx− cosfω0

radðt0 þ z0=cÞgy�

þ δ

�sinfϵω0

radðt0 þ z0=cÞgx− cosfϵω0

radðt0 þ z0=cÞgy��

ð32aÞ

B0axisðz0; t0Þ ¼ γ0Bu

��cosfω0

radðt0 þ z0=cÞgxþ sinfω0

radðt0 þ z0=cÞgy�

þ δ

�cosfϵω0

radðt0 þ z0=cÞgxþ sinfϵω0

radðt0 þ z0=cÞgy��

; ð32bÞ

where the definitions given in Eq. (5) hold. Moreover, thecomments following Eq. (5) and the subsequent equations(6), (7), and (8) are also valid for the circular polariza-tion case.Now we substitute Eq. (32) in Eq. (8):

dγ0mv0xdt0

¼ eγ0c

�1þ v0z

c

�Bu

�sin

�ω0rad

�t0 þ z0

c

��

þ δ sin

�ϵω0

rad

�t0 þ z0

c

���ð33aÞ

dγ0mv0ydt0

¼ −eγ0c�1þ v0z

c

�Bu

�cos

�ω0rad

�t0 þ z0

c

��

þ δ cos

�ϵω0

rad

�t0 þ z0

c

���ð33bÞ

dγ0mv0zdt0

¼ −eγ0Bu

�v0x

�sin

�ω0rad

�t0 þ z0

c

��

þ δ sin

�ϵω0

rad

�t0 þ z0

c

���

− v0y

�cos

�ω0rad

�t0 þ z0

c

��

þ δ cos

�ϵω0

rad

�t0 þ z0

c

����ð33cÞ

dγ0mc2

dt0¼ eγ0cBu

�v0x

�sin

�ω0rad

�t0 þ z0

c

��

þ δ sin

�ϵω0

rad

�t0 þ z0

c

���

− v0y

�cos

�ω0rad

�t0 þ z0

c

��

þ δ cos

�ϵω0

rad

�t0 þ z0

c

����; ð33dÞ

where

γ0 ≡ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − v02x þv02y þv02z

c2

q : ð34Þ

Multiplying Eq. (33c) by c and adding to Eq. (33d)yields the same constant of motion as in Eq. (11).

FIG. 4. Linear polarization: For a single electron the on-axisintensity of the first three odd harmonics vs K at a distance of100λu. The electron energy is 60 MeV.

FIG. 5. Linear polarization: Far-field spectral energy density(energy/wavelength), as measured along the x ¼ y plane, fortwo different values of K. The electron energy is 60 MeVwhile λu ¼ 1.39 cm.

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Equations (12), (13), and (14) also hold for the circularpolarization case. Moreover, just like the linear polarizationcase, we will make use of the definitions given in Eq. (15)along with the following:

u0y ≡ v0y=c ð35aÞ

ζ0 ≡ ω0rad

cy0 ¼ 2πγ

Aλuy0 ¼ ζ0ð0Þ þ

Zτ0

0

u0yðτ″Þdτ″: ð35bÞ

On one hand, using Eqs. (34) and (11), we get thefollowing normalized equation:

γ02ðu02x þ u02y Þ ¼ 2Aγ0 − ð1þ A2Þ: ð36Þ

On the other hand, consider the normalized version ofEqs. (33a) and (33b):

dγ0u0xdτ0

¼ Kð1þ u0zÞ½sinðτ0 þ η0Þ þ δ sinfϵðτ0 þ η0Þg�ð37aÞ

dγ0u0ydτ0

¼ −Kð1þ u0zÞ½cosðτ0 þ η0Þ þ δ cosfϵðτ0 þ η0Þg�:ð37bÞ

Equation (37) can exactly be integrated to yield

γ0u0x ¼ −K�cosðτ0 þ η0Þ þ δ

ϵcosfϵðτ0 þ η0Þg

�ð38aÞ

γ0u0y ¼ −K�sinðτ0 þ η0Þ þ δ

ϵsinfϵðτ0 þ η0Þg

�: ð38bÞ

Equation (38) implies

γ02ðu02x þu02y Þ ¼K2

�1þ

�δ

ϵ

�2

þ 2δ

ϵcosfð1− ϵÞðτ0 þ η0Þg

�:

ð39Þ

Now, comparing Eqs. (36) and (39) yields

γ0 ¼ ð1þ A2Þ þ K2½1þ ðδϵÞ2 þ 2 δϵ cosfð1 − ϵÞðτ0 þ η0Þg�

2A:

ð40Þ

Then consider the longitudinal momentum that can beobtained from Eqs. (40) and (11):

γ0u0z ¼A2 − 1 − K2½1þ ðδϵÞ2 þ 2 δ

ϵ cosfð1 − ϵÞðτ0 þ η0Þg�2A

:

ð41Þ

Since the average longitudinal momentum should bezero in the electron frame, we get the following usefulresult from Eq. (41):

A2 ¼ 1þ K2

�1þ

�δ

ϵ

�2�: ð42Þ

For the balanced hybrid HE1n modes, Λ ≈ 1 ⇒ δϵ ¼

1−Λ1þΛ ≈ 0 ⇒ A2 ¼ 1þ K2. Hence, Eq. (14) becomes

λrad ¼λu2γ2

½1þ K2 þ ðγθÞ2�: ð43Þ

Note that Eq. (43) is analogous to the correspondingequation for the circular polarized PMUs [16]. With theassumption of HE1n modes, we can simplify the relevantequations of motion as follows:

γ0 ¼ A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ K2

pð44aÞ

u0z ¼ 0 ð44bÞ

u0x ¼ −KAcos τ0 ð44cÞ

u0y ¼ −KAsin τ0 ð44dÞ

η0 ¼ 0 ð44eÞ

ξ0 ¼ 2πγ

λuAx0 ¼ −

KAsin τ0 ð44fÞ

ζ0 ¼ 2πγ

λuAy0 ¼ K

Acos τ0: ð44gÞ

It is clear from the last two equations that the electronrotates in a circle of radius K

2πγ λu in the transverse plane ofthe electron frame. As the transverse coordinates arepreserved under the Lorentz transformation to the labframe, this is also the radius of the helical motion of theelectron in the lab frame.Analogous to Eq. (28), we can now obtain motion

parameters in the lab frame for the circular polarization case:

ξ≡ 2π

λux ¼ A

γξ0 ¼ −

Kγsin τ0 ð45aÞ

ζ≡ 2π

λuy ¼ A

γζ0 ¼ K

γcos τ0 ð45bÞ

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η≡ 2π

λuz ¼ η0 þ τ0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

�Aγ

�2

s¼ τ0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

�Aγ

�2

sð45cÞ

τ≡ 2πcλu

t ¼ τ0 þ η0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

�Aγ

�2

s¼ τ0 ð45dÞ

ux ≡ vxc¼ Au0x

γh1þ u0z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðAγÞ2

q i ¼ −Kγcos τ0 ð45eÞ

uy ≡ vyc¼ Au0y

γh1þ u0z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðAγÞ2

q i ¼ −Kγsin τ0 ð45fÞ

uz ≡ vzc¼

u0z þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðAγÞ2

q1þ u0z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − ðAγÞ2

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 −

�Aγ

�2

sð45gÞ

αx ≡ λu2πc2

dvxdt

¼ duxdτ

¼ Kγsin τ0 ð45hÞ

αy ≡ λu2πc2

dvydt

¼ duydτ

¼ −Kγcos τ0 ð45iÞ

αz ≡ λu2πc2

dvzdt

¼ duzdτ

¼ 0: ð45jÞ

Now we can use Eqs. (30) and (31) to calculate the far-field radiation due to a single electron at a space-time pointðx0; y0; z0; ct0Þ≡ λu

2π ðξ0; ζ0; η0; τ0Þ in the lab frame.In Fig. 6, we show the intensity of various harmonics

versus the off-axis angle θ in a far-field transverse plane,

100λu ahead of the electron. The electron energy is chosento be 60 MeV. Except for the fundamental harmonicH1, allother harmonics are more intense off axis. As we go fromK ¼ 0.5 to K ¼ 1.0, the intensity of the fundamentalharmonic decreases while that of other harmonics increaseby an order of magnitude. Figure 7 plots the on-axisintensity of the fundamental harmonic against differentvalues of K. Note that the intensity of the fundamentalharmonic reaches maximum atK ≈ 0.57 and then decreasesas we increase K.As was done for the case of linear polarization, we can

calculate the spectrum of the radiation along a line in thefar-field transverse plane by calculating the intensity andthe wavelength of the fundamental harmonic along that

(a) (b)

(c) (d)

FIG. 6. Circular polarization: For a single electron the intensity (in units of 10−12 Wm2=λ4u) at a distance of 100λu of the first three oddand even harmonics versus the off-axis angle θ in a far-field transverse plane. The electron energy is 60 MeV.

FIG. 7. Circular polarization: For a single electron the on-axisintensity of the fundamental harmonic H1 versus K at a distanceof 100λu. The electron energy is 60 MeV.

MUHAMMAD SHUMAIL and SAMI G. TANTAWI PHYS. REV. ACCEL. BEAMS 19, 074001 (2016)

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line. According to Eq. (43) the wavelength increases as wego off axis. However, for the case of circular polarization,the orientation of the observation line in the far-fieldtransverse plane does not matter. Figure 8 shows thespectra calculated for K ¼ 0.5 (blue) and K ¼ 1.0(red, dashed). For these calculations, γ ¼ 117.417(corresponding to 60 MeV electron energy) andλu ¼ 1.39 cm. As in the case of linear polarization,the spectra shown in Fig. 8 are obtained by convolvingthe line function spectrum, corresponding to the durationof 78 cycles of undulation, with the correspondingcontinuous theoretical spectrum. Here again the peaksof the spectra lie a little right to the on-axis radiationwavelengths given by Eq. (43). These wavelengths aremarked by vertical dashed lines in Fig. 8.

V. CONCLUSION

We have derived the electromagnetic equivalent defini-tions of undulator period (λu) and undulator deflectionparameter (K) for linear and circular polarized dipolemodes: TM1n, TE1n, and HE1n. We have shown thatin the electron frame, where the average momentum ofthe electron is zero, it traces a figure-8-like trajectory for thelinear polarization mode and a circular trajectory for thecircular polarization mode. The parametric equations ofthese trajectories are derived. For balanced hybrid HE1nmodes, these trajectories are found similar to those in thecase of PMUs. The amplitude of the sinusoidal trajectory(linear polarization) or the radius of the helical trajectory(circular polarization) is determined as K

2πγ λu. We have alsoobtained the curves for the intensity of various harmonicsversus the off-axis angle. It turns out that for the linearpolarization the odd harmonics are stronger on the axiswhile the intensity of even harmonics is maximum at some

off-axis angle. For the circular polarization only thefundamental harmonic is present on the axis. ForK > 0.5, the intensity of higher harmonics increases atthe expense of that of the fundamental harmonic. Themaximum intensity of the fundamental harmonic occurs forK ≈ 0.75 and K ≈ 0.57 for the linear and circular polari-zation, respectively. It is also observed that the far-fieldintensity of harmonics is proportional to the sixth power ofelectron energy. Moreover, we have presented the theo-retical spectra of synchrotron radiation for various cases. Inprinciple, one can use the equations derived in this paper tocalculate the spectrum for any particular parameters of anelectromagnetic insertion device. The fundamental on-axisradiation wavelength can then be estimated by correlatingthis theoretical spectrum with the measured one, as we havedone during the data analysis of our microwave undulator[8]. It is hoped that the theoretical study presented in thispaper would be a useful benchmark for the tunable IDswhich are based on electromagnetic fields.

ACKNOWLEDGMENTS

The project was funded by the U.S. Department ofEnergy and DARPA AXiS program. M. S. is also thankfulto Habib University who provided MATHEMATICA

® soft-ware that was used for mathematical analysis and graphplotting in this paper.

APPENDIX: DOPPLER SHIFT AT RELATIVISTICSPEEDS AND NARROW ANGLES

According to Lorentz transformations, the frequency ω0of the wave in the electron frame is related to its frequencyω in the lab frame as follows:

ω0 ¼ γ0ωð1 − v0=c cos θÞ: ðA1Þ

This is the general relativistic Doppler-shift equation.For narrow angles (θ ≪ 1), we can approximate it as

ω0 ≈ γ0ω

�1 −

v0cþ v0

cθ2

2

�

≈ γ0ω1 − ðv0c Þ2 þ ð1þ v0

c Þ v0c θ2

2

1þ v0c

≈ ω1þ ð1þ v0

c Þ v0c ðγ0θÞ22

γ0ð1þ v0c Þ

: ðA2Þ

Moreover, under the assumption of relativistic speed(v0=c ≈ 1) and rearranging the terms, this equation furthersimplifies to

ω ≈ ω0 2γ01þ ðγ0θÞ2

: ðA3Þ

FIG. 8. Circular polarization: Far-field spectral energy density(energy/wavelength) for two different values of K. The electronenergy is 60 MeV while λu ¼ 1.39 cm.

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[1] The Science and Technology of Undulators andWigglers (Oxford University Press, New York, 2004),Chap. 1.

[2] H. Motz, Applications of the radiation from fast electronbeams, J. Appl. Phys. 22, 527 (1951).

[3] H. Motz, W. Thon, and R. N. Whitehurst, Experiments onradiation by fast electron beams, J. Appl. Phys. 24, 826(1953).

[4] P. Emma, R. Akre1, J. Arthur, R. Bionta, C. Bostedt, andJ. Bozek, First lasing and operation of an ångstrom-wavelength free-electron laser, Nat. Photonics 4, 641(2010).

[5] T. Ishikawa, H. Aoyagi, T. Asaka, Y. Asano, N. Azumi,and T. Bizen, A compact X-ray free-electron laseremitting in the sub-ångström region, Nat. Photonics 6,540 (2012).

[6] Synchrotron Radiation Instrumentation, edited by T.Warwick, J. Arthur, H. A. Padmore, and J. Stöhr, AIPConf. Proc. No. 705 (AIP, San Francisco, 2003).

[7] R. Pantell, G. Soncini, and H. Puthoff, Stimulated photon-electron scattering, IEEE J. Quantum Electron. 4, 905(1968).

[8] S. Tantawi, M. Shumail, J. Neilson, G. Bowden, C. Chang,E. Hemsing, and M. Dunning, Experimental demonstrationof a tunable microwave undulator, Phys. Rev. Lett. 112,164802 (2014).

[9] Particle Accelerator Physics (Springer, New York, 2007),Chap. 20.

[10] The Science and Technology of Undulators andWigglers (Oxford University Press, New York, 2004),Chap. 4.

[11] K. Batchelor, Microwave undulator, in Proceedings of theLinear Accelerator Conference, Stanford, California, 1986(SLAC, Stanford, 1986), pp. 272–275.

[12] C. Pellegrini, X-Band microwave undulators for shortwavelength free-electron lasers, AIP Conf. Proc. 807, 30(2006).

[13] M. Shumail, G. Bowden, C. Chang, J. Neilson, and S.Tantawi, Application of the balanced hybrid mode inovermoded corrugated waveguides to short wavelengthdynamic undulators, in Proceedings of the 2ndInternational Particle Accelerator Conference, SanSebastián, Spain (EPS-AG, Spain, 2011), pp. 3326–3328.

[14] Classical Electrodynamics (John Wiley & Sons, Inc.,New York, 1999), Chap. 11.

[15] Classical Electrodynamics (John Wiley & Sons, Inc.,New York, 1999), Chap. 12.

[16] Particle Accelerator Physics (Springer, New York, 2007),Chap. 21.

[17] Free-Electron Lasers in the Ultraviolet and X-ray Regime:Physical Principles, Experimental Results, TechnicalRealization (Springer, New York, 2014), Chap. 2.

[18] The Science and Technology of Undulators and Wigglers(Oxford University Press, New York, 2004), Chap. 3.

[19] Classical Electrodynamics (John Wiley & Sons, Inc.,New York, 1999), Chap. 14.

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