A double FEL oscillator: A possible scheme for a photon–photon collider

Nuclear Instruments and Methods in Physics Research B 309 (2013) 171–176

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A double FEL oscillator: A possible scheme for a photon–photon collider G. Dattoli, A. Torre ⇑ ENEA UTAPRAD–MAT Laboratorio di Modellistica Matematica, via E. Fermi 45, 00044 Frascati, Rome, Italy

a r t i c l e

i n f o

Article history: Received 13 February 2013 Received in revised form 15 March 2013 Available online 26 March 2013 Keywords: Free-electron laser Photon–photon collision Quantum vacuum

a b s t r a c t Exploration of the mutual scattering of photons in vacuum is considered as a fundamental test of the quantum electrodynamics theory. In this connection, we propose a ‘‘double’’ free-electron laser oscillator as a possible device for head-on photon–photon collisions. The device is conceived to comprise two undulator sections within the same cavity, where then two laser beams are produced by two counterpropagating electron beams. The latter are in turn exploited to produce gamma photons by backward Compton scattering of the intracavity FEL radiation itself. A preliminary analysis of the collision rate of the backscattered photons is presented specifically at the maximum of the relevant cross section. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction In contrast with the principle of superposition of the classical electromagnetic theory and the strict linearity of Maxwell’s equations for a free electromagnetic field, quantum electrodynamics (QED) predicts the direct scattering of two photons mediated by the fluctuating electron–positron pairs of the quantum vacuum [1–8]. Accordingly, the exploration of the mutual scattering of photons in vacuum is considered as fundamental within the general context of the quantum vacuum investigations [9]. However, it may be experimentally practicable only in the high-intensity regime owing to the small cross section of the process. The total cross section for unpolarized initial photons runp in the center-of-mass frame, where photons of equal energy collide, has different trends in the low-energy and high-energy regions with the photon energy Eph ¼  hxph being measured in units of the electron rest energy E0 ¼ me c2 ¼ 0:511 MeV [5–8]. In the low-energy region, roughly identified by the limit E ph  Eph =E0 ’ 0:5; runp increases with the photon energy/frequency through a sixth-power

rðlow-energyÞ ’ 0:13E 6ph lb, whilst in the unp which E ph > 10; runp decreases as lb. Finally, runp reaches its maximum value

dependence according to high-energy region, for

rðhigh-energyÞ ’ 20E 2 unp ph ðmaxÞ ðmaxÞ runp ’ 1:6  1030 cm2 at E ph ’ 1:5.

As a consequence of the small value of the scattering cross section, the photon–photon collider strategy grounds on the use of high-power lasers (see [10,11] for a review), among which the free-electron lasers (FELs) offer outstanding, if not unique, ⇑ Corresponding author. E-mail address: [email protected] (A. Torre). 0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.03.034

potential advantages [12]. In fact, as stated in [13,14], ‘‘modetarely upscaled versions of existing FELs would provide optimal conditions for the detection of scattering of light by light.’’ In the quoted papers, a possible experimental detection scheme was suggested to exploit the UV photon-beam provided by an FEL-amplifier; such a beam should be split into two pulses, each of which through a Compton back-scattering by the FEL driving electron-beam would produce c-photons for the desired head-on collision. As an alternative to the configuration, proposed in [13,14], we will propose here a possible FEL oscillator-based scheme for a photon–photon collider and investigate its feasibility at a primary level. The proposed scheme resorts to a compact structure, which, comprising two undulator sections within a single resonator, would allow for the production of two counterpropagating c-photon beams by backward Compton scattering of the intracavity FEL photons by the electron beams, which so will drive both the FEL oscillation and the photon–photon scattering. The head-on c–c collisions would occur inside the resonator as well. Notably, instead of taking advantage of the high peak power radiation, potentially deliverable by an FEL device (possibly through a multi-stage configuration as well), which is in contrast central to other FEL-based photon collider schemes [15], the device we propose resorts to the availability of electron-beam delivery systems, that could both operate at high-repetition rate and deliver ‘‘long’’ e-bunches, to produce photons in such a way that detection of their collision be experimentally accessible. Thus, in Section 2 we will briefly describe the double-undulator FEL oscillator, essentially highlighting the basic parameters that characterize the inherent performance. In Section 3 we will investigate the detectability of the photon–photon scattering, providing an estimate of the relevant collision rate for suitable ranges and/or

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specific values of the pertinent FEL parameters. Indeed, detection of the photon–photon scattering will be shown feasible even with a ‘‘comfortable’’ design of the proposed double FEL oscillator (and relevant electron-beam delivery system). Concluding remarks will be given in Section 4.

Moreover, in accord with well-known scaling laws [17], the equilibrium intracavity FEL radiation intensity IFEL can be given in the form

"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 1g IFEL ½W=cm  ¼ 2:41 Gmax ðg 0 Þ  1 IS ; 2

ð2Þ

g

2. A double FEL oscillator as a photon–photon collider: scheme and basic parameters The central part of the setup we propose for a photon–photon collider consists of a double-undulator based FEL oscillator, as schematized in Fig. 1. It is composed of two identical linear-undulator sections closed within an optical cavity, and driven by two independent counterpropagating, but perfectly synchronized, electron beams. In each undulator section, an e-beam is injected with a twofold objective: (i) to produce FEL radiation in the cavity within a suitable frequency range, and (ii)) to scatter FEL photons (through Compton back-scattering) into higher-energy photons (possibly c-photons) to facilitate the detection of photon–photon collisions, on account of the behavior of rðE ph Þ, as previously recalled. Evidently, the device is structured to have two conversion regions, conveniently located at the exit (relatively to the travelling electrons) of the undulators. There, through backward Comptonscattering of FEL photons by the incoming electrons, higher-frequency photons are produced for the desired head-on collision, that should occur centrally in the cavity. For a primary analysis of the collider performance, we ignore the detailed setup structure, reasoning as if we were considering a FEL oscillator composed of a single (linear) undulator, which as usual will be characterized by the relevant number of periods N U , the period length kU , and the undulator strength parameter K U , explicitly given as [16]

KU ¼

kU ½cmB0 ½kG ; 10:71

B0 being the on-axis undulator magnetic field. Then, as we know, the virtual undulator photons (with angular frequency xU ¼ 2kpUc) are back-scattered by the on-axis travelling electrons into the real FEL photons, whose central wavelength and angular frequency are

kFEL ¼

K 2U

kU 1þ 2c2e 2

! ;

xFEL ¼

2 e

2c xU ð1 þ K 2U =2Þ

with g denoting the cavity losses, Gmax ðg 0 Þ the FEL maximum gain in the small-signal small-gain regime (suitable to the oscillator operation), and IS the FEL oscillator saturation intensity. The former depend on the specific cavity design, whereas, resorting to the aforementioned scaling laws [17], Gmax ðg 0 Þ and IS can be written as

Gmax ðg 0 Þ ¼ 0:85g 0 þ 0:192g 20 þ 4:23  103 g 30   2:77  105 ce IS ½W=cm2  ¼ J ½A=cm2 ; Rðg 0 Þ NU e

ð1Þ

where ce ¼ Ee =E0 is the electron relativistic factor, Ee being the electron energy.

ð3Þ ð4Þ

with

Rðg 0 Þ ¼ g 0 ð1 þ 0:19g 0  8:7  103 g 20 þ 2:7  104 g 30 Þ

ðg 0 6 20Þ:

Here, J e ½A=cm2  is the electron-beam (peak) current density,

J e ½A=cm2  ¼

Ie ½A Ie ½A ; ¼ 2prx ½cmry ½cm 2Re ½cm2 

ð5Þ

which conforms to an inherent transverse dependence like 2

Jðx; yÞ ¼ J e e

y2

 x 2 2 2r 2r x y

;

rx ; ry signifying the beam sizes, respectively in the horizontal and vertical directions, defined by the pertinent variances of the beam particle density, so that the electron beam current follows as R Ie ½A ¼ Jðx; yÞdxdy. Also, as signalized in (5), the cross sectional area of the electron beam is taken as Re ½cm2   prx ry . Finally, g 0 is the FEL small-signal gain coefficient [16,17], g 0 ¼ 3:7  104 J e ½A=cm2 

 3 NU

ce

½kU ½cmK U F B ðnU Þ2 ;

ð6Þ

where F B ðnU Þ is the Bessel function factor,

F B ðnU Þ ¼ J 0 ðnU Þ  J 1 ðnU Þ; ;

ðg 0 6 10Þ;

nU 

K 2U 4ð1 þ 0:5K 2U Þ

J0 ; J 1 representing the Bessel functions of first order. It is worth remarking that, in accord with the limitations specified in (3), the expression (2) for IFEL holds for values of the gain

Fig. 1. ‘‘Double’’ FEL oscillator as a possible device for head-on photon–photon collisions. The device is structured to have two conversion regions, where, through Compton back-scattering of FEL photons by incoming electrons, higher-frequency photons are produced for the desired head-on collision, which should occur in the middle of the optical cavity.

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coefficient g 0 6 10, which condition, as said, identifies the smallgain regime. 3. Head-on collision of FEL intracavity Compton backscattered photons: an estimate of the relevant rate at the peak of runp The Compton back-scattering of electromagnetic radiation by electrons has been extensively discussed in the literature. It was also considered in connection with wave-undulator FELs, i.e. FEL devices based on real electromagnetic waves rather than on magnetic undulators (see, for instance, [18] for a recent analysis); we may recall indeed that, in the light of the Weizsaecker–Williams approximation, the magnetic-undulator FEL operation can be analysed in terms of the Compton back-scattering of the virtual undulator photons by the electrons. Likewise, the parameter that characterises the scattered radiation is the wave-undulator strength parameter, which, in fact, paralleling the magnetic-undulator strength parameter K U , in the case of concern here, is written as [18]

produce c-photons with quite ordinary electron-beam energies, i.e. ce P 150; for instance, photons with energies near the maximum of the scattering cross section are obtained with ce ’ 210—220. The plots in Fig. 2 show the behavior of N_ ph as a function of the ebeam parameters ce and J e , which can presumably be considered as more crucial than the other parameters, for which the typical values: kU ¼ 2:5 cm; N U ¼ 150, K U ¼ 0:5 and g ¼ 0:01 have been taken. As far as the values of the device parameters pertaining to the figure are concerned, it is evident that with increasing ce the rate of variation of N_ ph with J increases as well, so that the differences e

among the values of N_ ph corresponding to different values of J e become more evident for higher values of ce . Also, the interval of optimal values of c , over which indeed N_ ph is larger, shifts towards e

larger values of ce with increasing the beam density current. In fact, Table 1 conveys the values of J e and ce at which for the given set of parameters, kU ¼ 2:5 cm; N U ¼ 150, K U ¼ 0:5 and g ¼ 0:01; N_ ph

ð7Þ

takes on its maximum values, also reported in the Table. Interestingly, the involved values of J e and ce can be well represented by   the relation: c ðN_ ph  Þ ’ 9:035J 0:433 þ 41:034, which holds also

We restrict our analysis to the case that K FEL < 1, which evi-

for higher values of J e . We see that the range ce ¼ 20—160 seems to be suitable to produce low-energy photons (i.e. with E ph K 0:5), respectively

K 2FEL ¼ 7:36  1011 k2FEL ½cm2 IFEL ½W=cm2  ¼ 2:62  1012

IFEL ½W=cm2  : x2FEL ½s1 

dently amounts to

IFEL ½W=cm2  x2FEL ½s1 

e

< 0:4  1012 , under which condition

the scattering into higher-order harmonics can be neglected. Then, following the analysis in [13,14], the number of photons backCompton scattered per unit time by one electron within the cone of full aperture 2=ce around the direction of motion of the electron beams, i.e. the FEL cavity optical axis, turns out to be

IFEL ½W=cm2  N_ ph ½s1  ¼ axFEL K 2FEL FðK FEL Þ ¼ 1:91  1010 FðK FEL Þ; xFEL ½s1 

ð8Þ

where a ¼ 1=137:04 is the fine structure constant, and the function FðK FEL Þ is [13,14]

FðK FEL Þ ¼

8 þ 9K 2FEL =2 þ 3K 4FEL =4 3ð2 þ K 2FEL =2Þ3 ð1 þ K 2FEL =2Þ

:

ð9Þ

Then, the frequency of the scattered photons is

xph ¼ 4c2e xFEL ¼

4 e

8c xU ð1 þ K 2U =2Þ

;

e

max

yielding UV photons at ce ¼ 20 (E ph ’ 104 ) and hard X-rays photons at ce ¼ 160 (E ph ’ 0:5), whereas higher values of ce should convey c-photons, thus allowing one to reach the intermediate and high-energy regions along the photon-energy line inherent

rðE ph Þ.

to ðmaxÞ Eph

In

particular,

the

value

ðmaxÞ

E ph

’ 1:5

(i.e.

’ 0:77 MeV), corresponding to the peak of the photon-colliðmaxÞ

sion cross section runp ’ 1:6  1030 cm2 , is reached at ce ’ 216. It is worth noting that, in order for the values conveyed by the expression ( 8) to be physically meaningful, they must be positive. This amounts to the condition

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1g Gmax ðg 0 Þ > 1;

g

which, with the approximation Gmax ðg 0 Þ  0:85g 0 , yields the inequality

ð10Þ

with correspondingly Eph ¼  hxph ’ 8 hc4e xU . Notably, the undulator frequency xU is upscaled by the fourth power of the electron-beam energy, so that by xU s typically in the microwave range one can

3:15  104

1g

g

½kU K U F B ðnU Þ2

N 3U

c3e

J e > 1:

ð11Þ

Evidently, it can be solved for any one of the inherent parameters for fixed values of the other.

Fig. 2. N_ ph vs. (a) Je for ce ¼ 50 (solid red line), ce ¼ 100 (dotted blue line), ce ¼ 200 (dashed green line); and (b) ce for Je ¼ 150 A=cm2 (solid red line), Je ¼ 250 A=cm2 (dotted blue line), Je ¼ 350 A=cm2 (dashed green line), J e ¼ 450 A=cm2 (dot-dashed magenta line). The other parameters are set as the typical values: kU ¼ 2:5 cm; N U ¼ 150; K U ¼ 0:5 and g ¼ 0:01.

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Table 1 Maximum back-Compton scattering rate of photons   for kU ¼ 2:5 cm; N U ¼ 150; K U ¼ 0:5; g ¼ 0:01. N_ ph  max

by

one

electron

  N_ ph 

Je ½A=cm2 

ce

150

120

0:44  104

250

140

4

0:62  10

350

155

0:77  104

450

168

0:91  104

550

180

1:05  104

max

½s1 

parameters contribute merely through the multiplicative factor Re T 4e N b f . Note the fourth power-dependence on T e , which gives the micro-bunch length a significant role to determine the event detectability. The investigation of the specific behavior of the product N_ 2 J 2 r ph

E

ph

ðmaxÞ

The number of photon–photon collisions per unit time is given by the product of the luminosity L of the collider and the cross section r of the collision process:

N_ ½s1  ¼ Lr:

ð12Þ

Specifically, for the case of concern here, the luminosity L specializes as

value for r ¼ runp

Nc

4R

Nb f ½cm2 s1 ;

ð13Þ

where N c is the total number of photons produced by a single micro-bunch in the electron-beam and R is the sectional area, over which the scattering occurs; the latter is defined as R ¼ psx sy with sx ; sy signifying the relevant horizontal and vertical interaction region sizes. Also, N b denotes the number of micro-bunches in each e-bunch train that take part in the Compton back-scattering; in particular, it may equal the total number of micro-bunches in each ebunch train, if all the micro-bunches in the train are finalized to the photon production. Finally, f denotes the repetition rate of the electron beam delivery system, and hence the product N b f conveys the overall repetition rate of the scattering process. Evidently, N c can be expressed in terms of the number N_ ph of photons scattered per unit time by one electron according to

Nc ¼ F N_ ph Ne T e ;

ð14Þ

where N e gives the number of electrons in the micro-bunch, whose temporal length is T e ½s, so that

Ne ¼

2Re J e T e ; e

with e being the absolute value of the electron charge: e ¼ 1:602  1019 C. Finally, F  RRFEL is the filling factor, defined by e the ratio between the cross sectional areas of the FEL photon-beam and the e-beam. Therefore, on account of the explicit expression (8) for N_ ph ; N_ , can be written as

i2 N2c r h Nb f ¼ 2 N_ ph J e RFEL T 2e N b f N_ ½s1  ¼ r 4R e R 1:43  1058 r 2 2 2 4 2   J e RFEL IFEL T e F ðK FEL ÞNb f ; ¼ 2

xFEL

ð15Þ

ðmaxÞ

E ph

’ 1:5, amounting by (10) to

Eph ’ 8hc4e

xU ð1 þ K 2U =2Þ

which can be symplified under suitable hypotheses and in definite photon-energy regions. Indeed, under the reasonable assumption that

R  RFEL  Re ; the expression (15) symplifies into

N_ ½s1  ¼

 2 r _2 2 4 37 _ N N J R T N f ¼ 3:9  10 J rRe T 4e Nb f ; e b ph e ph e e 2

e

ð16Þ

whose behavior is ruled by the e-beam current density and energy, the undulator parameters and the cavity losses in a rather non trivial way basically through the product N_ 2ph J2e r, whereas the other

¼ 1:5E0 ;

yields a definite relation between ce and xU (or, kU ), given as

ce ¼ 167 k1=4 1þ U

K 2U 2

!1=4  cðmaxÞ : e

Specifically, the figures highlight the trend of CðJ e ; kU ; N U ; K U ; g), as a function of J e ; kU and N U (in Fig. 3, where K U ¼ 0:5 and g ¼ 0:01) and, in particular, as a function of Je and kU (in Fig. 4, where N U ¼ 150; K U ¼ 0:5 and g ¼ 0:01). The ranges of J e have been chosen in order to fulfill the condition (11) for the fixed values of kU and N U . Finally, we will give an estimate of the number of events, for inðmaxÞ stance, in one year N year for photons at E ph . Indeed, as conveyed by the values in Table 2, for J e ¼ 300½A=cm2 ; Re ’ 102 cm2 , and the already considered values kU ¼ 2:5 cm; N U ¼ 150; K U ¼ 0:5 and g ¼ 0:01, that yield xU ¼ 7:54  1010 Hz; ce ¼ 216, and hence xFEL ¼ 6:25  1015 ; Hz in the UV range, and N_ ph ¼ 0:62  104 s1 ,   one can get values for N year E ðmaxÞ , which range from ph  N year E ðmaxÞ ¼ 3:364  105 for T e ¼ 1 ns and N b f ¼ 5  105 Hz to ph N year  ðmaxÞ ¼ 6727  103 for T e ¼ 1 ls and N b f ¼ 105 Hz. E

ph

The values in Table 1 concretize the already remarked crucial dependence of N_ , on T e . Also, for the considered values one gets g 0 ¼ 0:055. In this connection, let us note that the value of g 0 determines that of Gmax ðg 0 Þ, which in turn conveys the relative average energy variation of the (monochromatic) electron beam as a consequence of the FEL operation along the undulator section. It is desiderable that such an energy variation be relatively small so that it could reasonably be neglected when dealing with the photon back-scattering by the electrons. To this end, from the request that g 0 K 0:2, implying that Gmax ðg 0 Þ ’ 0:85 g 0 K 0:2 as well, one can deduce, specifically ðmaxÞ for an e-beam energy ce ¼ ce of concern here, quite a safe condition for the basic FEL parameters J e and N U as

J e N3U K

R

’ 1:6 lb, only the term N_ 2ph J 2e in N_ 2ph J 2e r main-

tains the dependence on the device parameters; also, the condition

2

L¼

e

demands for the detailed account of the photon collision cross section rðE ph Þ as a function of E ph , and hence of ce ; kU and K U . Then, to fix mind, Figs. 3 and 4 show the behavior of the inherðmaxÞ : ent factor N_ 2ph J 2e , evaluated for r ¼ runp at E ph   2 2 N_ ph J e  ðmaxÞ  CðJ e ; kU ; N U ; K U ; gÞ. In that case, in fact, with the fixed

2:5  109 kU K 2U F B ðnU Þ2

:

Evidently, as seen, the values of J e and N U determine also the value of the equilibrium intracavity FEL radiation intensity IFEL , and hence of N_ ph and N_ . Therefore, the parameters should be chosen on the basis of an optimized balance between the various concurring effects. 4. Concluding notes We have proposed a compact two-stage photon–photon collider scheme, based on a double-undulator FEL oscillator. The device is conceived to produce two counterpropagating beams of

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175

Fig. 3. Plots of CðJe ; kU ; N U ; K U ; gÞ vs. J e for various values of kU , and N U ¼ 150 (solid red line), N U ¼ 200 (dotted blue line) and N U ¼ 250 (dashed green line). In all frames: K U ¼ 0:5 and g ¼ 0:01.

Fig. 4. (a) Plots of CðJe ; kU ; N U ; K U ; gÞ vs. Je for kU ¼ 2 cm (solid red line), kU ¼ 2:5 cm (dotted blue line), kU ¼ 3 cm (dashed green line), kU ¼ 3:5 cm (dot-dashed magenta line), and (b) relevant ðkU ; Je Þ-contourplots. In both frames: N U ¼ 150, K U ¼ 0:5 and g ¼ 0:01.

Table 2 ðmaxÞ Number of events in one year N year photons at E ph for kU ¼ 2:5 cm; ðmaxÞ N U ¼ 150; K U ¼ 0:5; g ¼ 0:01; Je ¼ 300½A=cm2 ; Re ’ 102 cm2 ; ce ¼ ce ¼ 216.  Te N b f (Hz) N year  ðmaxÞ E ph

1 ns

5  105

50 ns

5  105

3:364  105 210

100 ns

2:5  105

1682

1 ls

105

6727  103

primary photons by FEL oscillation in the designed cavity, and correspondingly two beams of c-photons, finalized to an head-on col-

lision inside the cavity as well. The latter result from the Compton back-scattering of the primary FEL–photons by the two counterpropagating electron-beams, injected into the resonator with the intent of driving both the FEL oscillation and the photon–photon scattering (Fig. 1). In practice, the device would then encompass both the photon-energy up-conversion and c–c collision regions within the same physical space. A preliminary analysis of the detectability of the photon–photon scattering has been presented, with the relevant collision rate being estimated for suitable ranges and/or specific values of the pertinent FEL parameters. As a result, the detection of the photon–photon scattering has been proven to be feasible even with a ‘‘comfortable’’

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design of the double FEL oscillator and relevant electron-beam delivery system. In fact, as said, resorting to the availability of electron-beam delivery devices, that could both operate at high-repetition rate and deliver ‘‘long’’ e-bunches, the proposed system can produce photons with energy near to the value at which the relevant cross section takes on its maximum value, thus making the detection of the collision process experimentally accessible. Also, the tunability of FEL radiation would allow for probing the scattering cross section vs. the photon energy over a certain range of the latter in proximity of the cross-section maximum. This probe would contribute as a test of the still untested QED predicted process of photon–photon scattering, which is considered as fundamental within the quantum vacuum investigations [10,11,9]. The analysis presented in the paper represents only a primary step; indeed, it grounds on simple scaling laws, which even though reliable and established on the basis of rigorous numerical simulations, cannot fully account for the complexity of the system dynamics. The aim of the present analysis is only that of establishing whether it makes sense to carry out a deeper investigation of the proposed device. As remarked, the obtained results lead one to rather optimistic conjectures. Several are, however, the issues to deal with, that will be the object of future presentations. We may mention for instance, the resonator diffraction losses, whose analysis (in view of minimizing them) demands, among others, for an accurate design of the cavity, or the influence of the cross-sectional and angular distribution of the electron beam on the overall system performance. Acknowledgement The authors thank the referees for their interesting comments and suggestions, that have improved the presentation.

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