Available via license: CC BY 4.0

Content may be subject to copyright.

Crab crossing in inverse Compton scattering

A. P. Potylitsyn,1,2 D. V. Gavrilenko ,2M. N. Strikhanov,2and A. A. Tishchenko 2,3,*

1National Research Tomsk Polytechnic University, Tomsk 634050, Russian Federation

2National Research Nuclear University “MEPhI”, Moscow 115409, Russian Federation

3National Research University “BelSU”, Belgorod 308034, Russian Federation

(Received 24 September 2022; accepted 6 March 2023; published 4 April 2023)

Inverse Compton scattering is a promising x-ray source, very bright, quasi-monochromatic, and

compact. In this paper, we present a generalized theory of Compton backscattering in terms of luminosity,

suitable for both classical and quantum regimes. We show that the optimal parameters, which require a

certain mutual orientation and inclination of the fronts of the laser and electron beams described by 3D

Gaussians, correspond to the crab scheme. This scheme is widely used in particle physics but is not yet used

for x-ray sources. The constructed theory not only predicts the optimal geometry for laser and electron

beams but also describes the luminosity. Our results reveal the opportunity to sharply increase the

luminosity of compact x-ray sources based on Compton/Thomson backscattering.

DOI: 10.1103/PhysRevAccelBeams.26.040701

I. INTRODUCTION

Inverse Compton scattering from relativistic electrons is

one of the most prospective ways to generate quasi-

monochromatic x-ray beams [1–9]. It occurs when photons

of a laser pulse are scattered on relativistic electrons so that

the maximal energy of scattered photons is proportional to

the squared Lorentz factor.

To obtain a high-intensity x-ray beam, the most simple

and direct way is not to increase the laser power (this leads

to a decrease in the monochromaticity of the x-ray beam in

a nonlinear regime), but to tilt the fronts of electron and

laser beams so that they interact for the longest time. This

requires the beams to be oriented along the approach

velocity, see Fig. 1. This idea was proposed by Palmer

in 1988 [10] for linear colliders and called the “crab-

crossing scheme”; then it was developed for ring-shaped

machines [11], and now is generally accepted in almost

all large accelerator/colliding facilities under the kindred

names “crab cavity, crab waist scheme”: the world record in

luminosity in the particle physics at KEKB.

High Luminosity LHC [12] means tenfold times increas-

ing of luminosity in 2029; proton-proton stage for Future

Circular Collider, positron-electron collider [13], future

Electron-Ion Collider being constructed jointly by Jefferson

Lab and BNL [14], etc.—they all are based on crab

crossing [15,16].

In terms of inverse Compton scattering, the first steps

in the direction of the realization of the crab-crossing

scheme were made in the papers [17,18].In[17], Bulyak

and Skomorokhov, using the luminosity representation,

showed that the maximal yield of scattered photons takes

place for a head-on collision of the electron and laser

beams. The theory constructed in terms of luminosity is

attractive as it allows one to take into account the effects

of not only classical but also quantum electrodynamics,

including nonlinear effects in strong laser fields, etc. Yet,

as the authors of [17] indicated, in the Compton sources,

the head-on collision is not acceptable for technological

reasons. In [18], Variola et al. elaborated the theory of [17]

considering the tilted electron beam to realize the crab-

crossing scheme. In [18], however, they did not consider

the tilt of the laser beam front, while, as one can see in

Fig. 1, or from the pioneer paper of Palmer [10],inideal,

the true crab-crossing scheme requires both beams to be

tilted.

In this study, we have generalized the theory developed

in [17,18]. Having accounted for the arbitrary angle of

crossing the electron and photon pulses and arbitrary

orientations of their fronts, we derive the closed analytical

expressions describing the luminosity. In Sec. IV, we show

that the influence of nonzero tilt of the laser front is vital for

the realization of optimal conditions that prove to coincide

with the crab-crossing scheme.

II. COLLISION OF ELECTRON AND PHOTON

BEAMS WITH DIRECT FRONTS

In quantum electrodynamics, the process of inverse

Compton scattering is described in a more universal way

than in classical electrodynamics, based on the cross

*tishchenko@mephi.ru

Published by the American Physical Society under the terms of

the Creative Commons Attribution 4.0 International license.

Further distribution of this work must maintain attribution to

the author(s) and the published article’s title, journal citation,

and DOI.

PHYSICAL REVIEW ACCELERATORS AND BEAMS 26, 040701 (2023)

2469-9888=23=26(4)=040701(7) 040701-1 Published by the American Physical Society

section and the luminosity of the process [19]. Using the

normalized 3D distribution FLðx; y; zÞof photons in a laser

pulse, and introducing the analogous distribution for

electron bunch Feðx; y; zÞ, one obtains the number of

scattered photons Nph:

Nph ¼NLNeσL;

L¼cð1þβcos φÞZdxdydzdtFLðx; y −ct; zÞ

×Feðx; y þβct; zÞ;ð1Þ

where NLand Neare the total numbers of particles in

the photon pulse and the electron bunch, L—the luminos-

ity, σ—the total cross section of the inverse Compton

scattering process, βc—the speed of the electron bunch,

cð1þβcos φÞis the relative speed of approach of the

beams (Fig. 2).

The distribution of laser photons in a form of the

production of three Gaussians is defined in the laboratory

coordinate system rotated by an angle φwith respect to the

dashed one:

FLðx; y; z; tÞ¼ 1

ð2πÞ3=2σLxσLy σLz

exp−

1

2 x0

σLx2

þy0−ct

σLy 2

þz0

σLz2;ð2Þ

where the dashed coordinates, corresponding to the system

rotated by an angle φ:

x0¼xcos φ−ysin φ;

y0¼xsin φþycos φ;

z0¼z: ð3Þ

After the substitution of Eq. (3) into Eq. (2), in the

approximation β¼1, the luminosity is calculated as

Lφ¼n2πﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

σ2

ez þσ2

Lz

q

×ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

σ2

ex þσ2

Lx þðσ2

ey þσ2

LyÞtan2ðφ=2Þ

qo−1:ð4Þ

Note that Eq. (4) equals zero at φ¼π. Therefore, the

sharp increase in the luminosity for φ→πshown in Fig. 3

of [17] appears to be incorrect. The general result obtained

in Sec. III confirms this conclusion, see Eq. (16) below.

III. COLLISION OF INCLINED ELECTRON

AND PHOTON BEAMS WITH TILTED FRONTS

(CRAB-CROSSING GEOMETRY)

For geometry shown in Fig. 3, the characteristics of

scattered photons will depend on three angular variables:

the angles of inclination for each of the beams (ξfor the

laser pulse and θfor the electron beam) and the angle of

noncollinearity φ—the angle between trajectories. To build

the theory that will let us consider the crab-crossing

geometry, we have to consider the collision of inclined

beams (arbitrary φ) having tilted fronts (arbitrary ξ,θ).

Considering the noncollinear geometry of collision

(Fig. 2), Bulyak and Skomorokhov suggested the method

that simplifies considerably calculation of luminosity [17].

Following them, we will use 3D Gaussian in the coordinate

frame x00,y00

FIG. 2. The layout of the process of inverse Compton scatter-

ing; both photon (left, orange) and electron beams (right, blue)

have direct fronts: the fronts are perpendicular to the trajectories.

FIG. 3. The layout of the process of inverse Compton scatter-

ing; both photon (left, orange) and electron beams (right, blue)

have tilted fronts.

FIG. 1. An optimal geometry—crab-crossing scheme: two

colliding beams are oriented along the velocity of approach

(dashed line).

A. P. POTYLITSYN et al. PHYS. REV. ACCEL. BEAMS 26, 040701 (2023)

040701-2

x00 ¼x0cos ξ−y0sin ξ;

y00 ¼x0sin ξþy0cos ξ;

z00 ¼z0ð5Þ

to describe the laser pulse with the front tilted under the

angle ξwith respect to the direction of propagation of the

pulse. After the substitution of Eq. (5) into the initial 3D

distribution, it is necessary to replace y0→y0−ct as in

Eq. (2) and only after that to pass on to the nondashed

coordinates with respect to Eq. (3).

By analogy, to consider the distribution of electrons in

the electron beam with front tilted under the angle θ

(Fig. 3), we use the replacement

x000 ¼xcos θ−ðyþβctÞsin θ;

y000 ¼xsin θþðyþβctÞcos θ;

z000 ¼z: ð6Þ

After all the substitutions into Eq. (1), the argument of the

exponent in the integrand will be a quadratic form in all

four variables. For simplicity, replacing the variables

x; y; z; ct →x1;x

2;x

3;x

4;ð7Þ

we write down this exponent as

exp−

1

2X

4

i;j

aijxixj:ð8Þ

Cumbersome calculations allow one to obtain the coef-

ficients aij in Eq. (8) depending on three angles and six

parameters of the Gaussians as elements of the symmetrical

matrix:

a11 ¼cos2ðθÞ

σ2

ex þsin2ðθÞ

σ2

ey þcos2ðξþφÞ

σ2

Lx þsin2ðξþφÞ

σ2

Ly

a22 ¼sin2ðθÞ

σ2

ex þcos2ðθÞ

σ2

ey þsin2ðξþφÞ

σ2

Lx þcos2ðξþφÞ

σ2

Ly

a33 ¼1

σ2

ez þ1

σ2

Lz

a44 ¼1

2β2

σ2

ex þβ2

σ2

ey þ1

σ2

Lx þ1

σ2

Ly þβ21

σ2

ey

−

1

σ2

excosð2θÞþ1

σ2

Ly

−

1

σ2

Lxcosð2ξÞ

a12 ¼1

2 1

σ2

ex

−

1

σ2

eysinð2θÞþ1

σ2

Lx

−

1

σ2

Lysin½2ðξþφÞ

a14 ¼1

2β1

σ2

ey

−

1

σ2

exsinð2θÞ−1

σ2

Ly þ1

σ2

LxsinðφÞþ1

σ2

Lx

−

1

σ2

Lysinð2ξþφÞ

a24 ¼1

2−β1

σ2

ex

−

1

σ2

eyþβ1

σ2

ex

−

1

σ2

eycosð2θÞþ1

σ2

Lx þ1

σ2

LycosðφÞþ1

σ2

Ly

−

1

σ2

Lxcosð2ξþφÞ;ð9Þ

The rest coefficients are zero. For a direct (untilted, ξ¼0)

electron beam and laser beam having a tilted front (θis

arbitrary), the coefficients aij from Eq. (9) coincide exactly

with those from the paper [18].

When the matrix ˆ

ais reduced to a diagonal form, the

exponent in Eq. (8) is written as

exp−

1

2X

4

i¼1

Aiη2

i:ð10Þ

Calculating the luminosity and using the replacement

fxig→fηigwith a unit Jacobian of the transition, instead

of a fourfold integral, it is required to calculate the

production of four single integrals:

L¼1

ð2πÞ3

1þcos φ

σexσey σezσLx σLy σLz Y

4

i¼1Zdηiexp−

Ai

2η2

i:

ð11Þ

Having performed the elementary integrating (the limits of

change of the variables ηirest infinite), we obtain

L¼1

2π

1þcos φ

σexσey σezσLx σLyσLz

1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det A

p:ð12Þ

CRAB CROSSING IN INVERSE COMPTON SCATTERING PHYS. REV. ACCEL. BEAMS 26, 040701 (2023)

040701-3

Thus, the main obstacle here is the calculation of the

determinant in the denominator. Having used the equality

det A¼det a, we obtain

det A¼1

σ2

ez þ1

σ2

Lz×½βcos θþcosðφ−θÞ2

σ2

eyσ2

Lxσ2

Ly

þ½βsin θ−sinðφ−θÞ2

σ2

eyσ2

Lxσ2

Ly þ½cos ξþβcosðφþξÞ2

σ2

eyσ2

Lxσ2

Ly

þ½sinðξÞþβsinðφþξÞ2

σ2

eyσ2

Lxσ2

Ly :ð13Þ

Following the designations from [18], one can present the

luminosity in Eq. (12) as

Lðφ;θ;ξÞ¼ 1

2πﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

σ2

ez þσ2

Lz

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

feðφ;θÞþfLðφ;ξÞ

p

;ð14Þ

feðφ;θÞ¼σ2

exβcos θþcosðφ−θÞ

1þβcos φ2

þσ2

eyβsin θ−sinðφ−θÞ

1þβcos φ2

;ð15Þ

fLðφ;ξÞ¼σ2

Lxcos ξþβcosðφþξÞ

1þβcos φ2

þσ2

LysinðξÞþβsinðφþξÞ

1þβcos φ2

:ð16Þ

The function feðφ;θÞcharacterizing the inclined electron

bunch coincides with that from [18] (see Eq. (11a) in [18]).

Equation (14) transforms into Eq. (10) from [18] at ξ¼0

(in [18] only case, ξ¼0was considered).

Note that Eqs. (15) and (16) depend on the inclination

angles ξand θthe same way (up to the direction), as one

would expect. Moreover, each of Eqs. (15) and (16) is

invariant relatively to the simultaneous replacement (σey ↔

σex,σLy ↔σLx ) and (θ↔θþπ=2,ξ↔ξþπ=2). Indeed,

the first pair of replacements change the axes, while the

second one returns them with the corresponding rotation.

IV. RESULTS AND DISCUSSION

To define the optimal angles corresponding to the

maximal luminosity, one has to differentiate Eq. (15):

∂fe

∂θ¼2

ð1þβcos φÞ2ðσ2

ex −σ2

eyÞΦeðφ;θÞΨeðφ;θÞ

∂2fe

∂θ2¼2

ð1þβcos φÞ2ðσ2

ex −σ2

eyÞfΨ2

eðφ;θÞ−Φ2

eðφ;θÞg

ð17Þ

Φeðφ;θÞ¼βcos θþcosðφ−θÞ

Ψeðφ;θÞ¼−βsin θþsinðφ−θÞð18Þ

Similarly, from Eq. (16), one obtains

∂fL

∂ξ¼2

ð1þβcos φÞ2ðσ2

Ly −σ2

LxÞΦLðφ;−ξÞΨLðφ;−ξÞ

∂2fL

∂ξ2¼2

ð1þβcos φÞ2ðσ2

Lx −σ2

LyÞ

×fΨ2

Lðφ;−ξÞ−Φ2

Lðφ;−ξÞg

ΦLðφ;ξÞ¼cos ξþβ× cosðφ−ξÞ;

ΨLðφ;ξÞ¼−sin ξþβsinðφ−ξÞ:ð19Þ

Thus, Eq. (14) is maximal when

σ2

ey >σ2

ex ⇒tan θoptimal ¼sin φ

βþcos φ;tan ξoptimal ¼−

sin φ

β−1þcos φ;

σ2

ey <σ2

ex ⇒tan θoptimal ¼−

βþcos φ

sin φ;tan ξoptimal ¼β−1þcos φ

sin φ:ð20Þ

Note that Eq. (20) defines the optimal angles θoptimal and ξoptimal at which the luminosity is maximal; and although these

optimal angles do not depend on the beam’s sizes, the optimalor maximal luminosity does, see Eq. (14).

Within the approximation β¼1, Eq. (20) reads

σ2

ey >σ2

ex ⇒θoptimal ¼φ=2;ξoptimal ¼−φ=2;

σ2

ey <σ2

ex ⇒θoptimal ¼φ=2þπ=2;ξoptimal ¼−φ=2þπ=2:ð21Þ

Of these conditions, the condition ξoptimal ¼−

φ

2was

obtained in [1] from geometrical optics by Debus et al.,

who were the first to propose the idea of using a laser pulse

with a tilted front to increase the effective length of

interaction between electrons and photons (the size of

the electron beam was neglected, and hence the angle θwas

not involved); the condition θoptimal ¼φ=2was obtained in

[18] in terms of luminosity representation.

A. P. POTYLITSYN et al. PHYS. REV. ACCEL. BEAMS 26, 040701 (2023)

040701-4

The reason why Eq. (21) contains two conditions

different for the beams of different forms is understandable:

the luminosity is maximal when the beams go through each

other being parallel to the direction along their “elongation”

(see Fig. 4and also Fig. 1), and a change in the beam

shapes can lead to a rotation of this direction up to π=2.

Thus, the maxima of elongated beams correspond to the

minima of the short beams and vice versa. As the elongated

beams are optimal, below we will consider this very case.

In Figs. 4(b) and 4(c), we demonstrate that when two

chains containing Nparticles each (the limiting case of

elongated beams) collide, the effective number of colliding

particles depends on the orientation of the chains: N2if they

are parallel to the velocity [Fig. 4(b)], and Nif they are

perpendicular [Fig. 4(c)]. Thus, in general case, when the

number of particles in longitudinal and transversal direc-

tions is NLand NTr correspondingly, then the effective

number of particles’collisions is N2

LNTr. Therefore, in

general case, it is advantageous for the elongated beams to

be oriented along the approach velocity, which coincides

with the crab-crossing scheme, see Fig. 1.

Note that a simple overlap of two beams in the collision

point is not sufficient: the key factor here is that both beams

while overlapping move along the directions of their

elongation. For example, both situations in Figs. 4(b)

and 4(c) imply a complete overlap of two beams, but only

Fig. 4(b) provides the optimal conditions of crab crossing.

It is evident therefore that the attempt to realize the crab-

crossing scheme undertaken in [18] without nonzero angle

ξcould not lead to the optimal conditions; below we will

see what role the angle ξplays.

Now let us analyze the dependence of the luminosity

from Eq. (14) on the characteristics of the laser pulse,

neglecting the dependence on the size of the electron beam

(σe→0). For simplicity, we consider the symmetric form

of the laser pulse, i.e., σLx ¼σLz ¼σL. Then Eq. (14) reads

Lðφ;0Þ¼ 1

2πσ2

Lﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1þðσLy=σLÞ2ðsin φ

1þcos φÞ2

q:ð22Þ

FIG. 4. Collision of the beams of different shapes: (a) general

case (the beams have finite thickness and length), (b) elongated

beams (each particle interacts with each: N2collisions), (c) short

beams (each particle interacts with another: Ncollisions). For

simplicity, the beams are depicted as having identical forms or

sizes.

FIG. 5. Dependence of the luminosity on the noncollinearity angle φfor in the case of scattering of the laser pulse on the electron

bunch with negligible sizes (σe→0) for various lengths of the pulse σLy : (a)—left—the laser pulse has perpendicular front (ξ¼0),

(b)—right—the front of the laser pulse is tilted, ξ¼−φ=2(optimal angle ξ).

CRAB CROSSING IN INVERSE COMPTON SCATTERING PHYS. REV. ACCEL. BEAMS 26, 040701 (2023)

040701-5

In Fig. 5(a), we show the dependences of the luminosity

on the noncollinearity angle φfor the standard pulse with

perpendicular front (ξ¼0) for various lengths of the

pulse σLy.

As follows from Fig. 5(a), for a short laser pulse,

noncollinearity (φ≠0) practically does not reduce the

luminosity compared with the head-on collision (φ¼0).

On the other hand, with increasing the pulse length, in the

case of σLy >σL, noncollinearity leads to the significant

suppression of the luminosity.

Yet, for the laser pulse with a front tilted at the angle

ξ¼−φ=2, the luminosity does not depend on the pulse

length and is defined by the transverse size only, see Fig. 5(b):

L1ðφ;ξ¼−φ=2Þ¼cosðφ=2Þ

2πσ2

L

:ð23Þ

Naturally, for a spherically symmetric pulse σLy ¼σL,the

luminosity does not depend on the angle ξ.

For a long laser pulse (σLy=σL¼5), the luminosity has a

maximum if ξ¼−φ=2(see Fig. 6). The authors of [18]

considered the laser beam with a perpendicular front ξ¼0;

the luminosity, in this case, is maximal only at φ¼0,

otherwise it is reduced, see Figs. 5and 6.

Let us consider the geometry when both beams, electron

and photon, have tilted fronts. To analyze the influence

of the sizes of both tilted beams on the luminosity, we

consider the ratio called the geometric factor in [18]:

Gðϕ;θ;ξÞ¼Lðϕ;θ;ξÞ=Lð0;0;0Þð24Þ

In Fig. 7, the geometric factor Gðπ=10;π=20;ξÞas a

function of the angle ξis shown for noncollinear collision

at φ¼π=10 and θ¼π=20. As follows from the figure,

there is a maximum at ξ¼−φ=2.

Note that results of [18], being obtained for the case of

untilted laser fronts only, correspond to the vertical line

ξ¼0in ξ-dependences in Figs. 6and 7. We see that the

accounting for the nonzero angle ξis of primary importance

for reaching a maximum radiation intensity.

VI. CONCLUSION

In this paper, we described colliding bunches by three-

dimensional Gaussians, which is more realistic than the

classical approach [1,2]. To consider the parameters of

colliding bunches in the next approximation depending on

the Rayleigh length for the laser and on the beta function

for the electron beam, it is necessary to simulate numeri-

cally the process of inverse Compton scattering [4,6,20].

Yet, the general theory developed above for an arbitrary

geometry, including all conceivable angles of mutual

orientation and inclination of the fronts of the laser and

electron beams, predicts the optimal geometry, expressed

by Eq. (20) (which corresponds to the crab-crossing

scheme well known in particle physics), and gives ana-

lytical formulas for the luminosity and the number of

scattered photons based on Eqs. (14)–(16), which are

applicable in a wide range of parameters, both within

the frames of classical electrodynamics (moderate laser

fields and linear effects) and quantum electrodynamics

(intense laser fields and nonlinear effects).

FIG. 6. The luminosity depending on the angle of tilt ξof the

front of a laser pulse for the long pulse (σLy=σL¼5).

FIG. 7. Geometric factor Gðπ=10;π=20;ξÞdepending on the

inclination angle of the laser pulse ξfor various ratios of its

longitudinal and transversal sizes when colliding with an inclined

electron bunch.

A. P. POTYLITSYN et al. PHYS. REV. ACCEL. BEAMS 26, 040701 (2023)

040701-6

On the other hand, the theory developed above is based

on the description of the process in terms of luminosity.

Using Eq. (1) only gives the total number of scattered

photons and does not provide any information about the

energy spectrum, divergence, and energy-angle correlation

of scattered photons. The spectral and angular distributions

of radiated or scattered photons, however, are possible

to obtain using a differential cross section instead of

the integral one in Eq. (1), see, e.g., [21]. Yet the quick

evaluation of the integral number of scattered photons as a

function of parameters of colliding beams is useful to

design Compton sources.

Allowing for the possibility of mutual rotation of laser

and electron pulses is the key to implementing the crab

scheme. As in particle and accelerator physics, this scheme

leads to a sharp increase in luminosity, we may hope that

the theory developed above can pave the way for a more

efficient x-ray source based on inverse Compton scattering.

ACKNOWLEDGMENTS

The study is supported by RFBR, Grant No. 19-29-

12036 (Sec. III), and by the Ministry of Science and Higher

education of the Russian Federation, the agreement

No. FZWG-2020-0032 (2019-1569) (Sec. II) and the

Contract No. 075-15-2021-1361 (Sec. IV). The contribu-

tion from A. P. P. was partly supported by the TPU (Grant

No. FSWW-2020-0008).

[1] A. D. Debus, M. Bussmann, M. Siebold, A. Jochmann, U.

Schramm, T. E. Cowan, and R. Sauerbrey, Traveling-wave

Thomson scattering and optical undulators for high yield

EUV and X-ray sources, Appl. Phys. B 100, 61 (2010).

[2] K. Steiniger, D. Albach, M. Bussmann, M. Loeser, R.

Pausch, F. Röser, U. Schramm, M. Siebold, and A. Debus,

Building an optical free-electron laser in the traveling-wave

Thomson-scattering geometry, Front. Phys. 6, 155 (2019).

[3] R. Rullhusen, X. Artru, and P. Dhez, Novel Radiation

Sources Using Relativistic Electrons (World Scientific,

Singapore, 1998).

[4] F. V. Hartemann, W. J. Brown, D. J. Gibson, S. G.

Anderson, A. M. Tremaine, P. T. Springer, A. J.

Wootton, E. P. Hartouni, and C. P. J. Barty, High-energy

scaling of Compton scattering light sources, Phys. Rev. ST

Accel. Beams 8, 100702 (2005).

[5] K. Deitrick, G. H. Hoffstaetter, C. Franck, B. D. Muratori,

P. H. Williams, G. A. Krafft, B. Terzić, J. Crone, and H.

Owen, Intense monochromatic photons above 100 keV

from an inverse Compton source, Phys. Rev. Accel. Beams

24, 050701 (2021).

[6] G. Paternò, P. Cardarelli, M. Bianchini, A. Taibi, I. Drebot,

V. Petrillo, and R. Hajima, Generation of primary photons

through inverse Compton scattering using a Monte Carlo

simulation code, Phys. Rev. Accel. Beams 25, 084601

(2022).

[7] S. G. Rykovanov, C. G. R. Geddes, C. B. Schroeder, E.

Esarey, and W. P. Leemans, Controlling the spectral shape

of nonlinear Thomson scattering with proper laser chirp-

ing, Phys. Rev. Accel. Beams 19, 030701 (2016).

[8] M. A. Valialshchikov, V. Yu. Kharin, and S. G. Rykovanov,

Narrow Bandwidth Gamma Comb from Nonlinear Comp-

ton Scattering Using the Polarization Gating Technique,

Phys. Rev. Lett. 126, 194801 (2021).

[9] A. Gonoskov, T. G. Blackburn, M. Marklund, and S. S.

Bulanov, Charged particle motion and radiation in strong

electromagnetic fields, Rev. Mod. Phys. 94, 045001

(2022).

[10] R. B. Palmer, Energy scaling, crab crossing and the pair

problem, in Proceedings of the 4th DPF Summer Study on

High-energy Physics in the 1990s (Snowmass, CO, USA,

1988), pp. 613–619, https://cds.cern.ch/record/194990?

ln=en.

[11] K. Oide and K. Yokoya, Beam-beam collision scheme for

storage-ring colliders, Phys. Rev. A 40, 315 (1989).

[12] The High Luminosity Large Hadron Collider, edited by O.

Brüning and L. Rossi (World Scientific, Singapore, 2015).

[13] W. Xu, J. Fite, D. Holmes, Z. A. Conway, R. A. Rimmer, S.

Seberg, K. Smith, and A. Zaltsman, Broadband high power

rf window design for the BNL Electron Ion Collider, Phys.

Rev. Accel. Beams 25, 061001 (2022).

[14] M. Zobov, D. Alesini, M. E. Biagini et al., Test of “Crab-

Waist”Collisions at the DAΦNE ΦFactory, Phys. Rev.

Lett. 104, 174801 (2010).

[15] Q. Wu, Crab cavities: Past, present, and future of a

challenging device, in Proceedings of IPAC 2015, 3643

(Richmond, VA, USA, 2015), THXB2.

[16] CERN, official website, https://cerncourier.com/a/crab-

cavities-enter-next-phase/.

[17] E. Bulyak and V. Skomorokhov, Parameters of Compton

x-ray beams: Total yield and pulse duration, Phys. Rev. ST

Accel. Beams 8, 030703 (2005).

[18] A. Variola, F. Zomer, E. Bulyak, P. Gladkikh, V.

Skomorokhov, T. Omori, and J. Urakawa, Luminosity

optimization schemes in Compton experiments based on

Fabry-Perot optical resonators, Phys. Rev. ST Accel.

Beams 14, 031001 (2011).

[19] J. Yang, M. Washio, A. Endo, and T. Hori, Evaluation of

femtosecond X-rays produced by Thomson scattering

under linear and nonlinear interactions between a low-

emittance electron beam and an intense polarized laser

light, Nucl. Instrum. Methods Phys. Res., Sect. A 428, 556

(1999).

[20] V. Petrillo, A. Bacci, R. Ben Alì Zinati, I. Chaikovska, C.

Curatolo, M. Ferrario, C. Maroli, C. Ronsivalle, A. R.

Rossi, L. Serafini, P. Tomassini, C. Vaccarezza, and A.

Variola, Photon flux and spectrum of g-rays Compton

sources, Nucl. Instrum. Methods Phys. Res., Sect. A 693,

109 (2012).

[21] P. Moskal, R. Czyżykiewicz (COSY-11 Collaboration),

Luminosity determination for the quasi-free nuclear reac-

tions, AIP Conf. Proc. 950, 118 (2007).

CRAB CROSSING IN INVERSE COMPTON SCATTERING PHYS. REV. ACCEL. BEAMS 26, 040701 (2023)

040701-7