Angular distribution of coherent Cherenkov radiation from a tilted bunch passing through a slit in target

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Πλούτων Αγγελόπουλος
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1 CHANNELING 016 Tomsk Polytechnic University, 30 Lenin Ave., Tomsk, Russia A.Potylitsyn, S. Gogolev Angular distribution of coherent Cherenkov radiation from a tilted bunch passing through a slit in target

the ChR wavelength) ChR can be produced.

2 The Cherenkov mechanism may be realized for charge passing in vacuum near a dielectric target. For the high Lorentz-factor γ and if the condition h γλ is fulfilled (h is the impact parameter, λ is the ChR wavelength) ChR can be produced. a) b) ch сos ch = β Fresnel s law sin = ε sin ch 1 ε 1 = arccos β ε If ε < γ ( ( )) = arcsin ε sin ch then the geometry a) can be realized ( ( )) = 90 α arcsin ε cos α + ch

3 3 Experiments T. Takahashi Physical Review E., v.6 (000) 8606 impact-parameter h=5 mm The schematic view in the vacuum chamber. (a) The sectional diagram of the optical components, (b) the block of quartz, and (c) the cone of Teflon with the cylindrical hole of 7 mm. (M6, M7, M8) plane mirrors; (M9) a spherical mirror; and ( ) electron beam. The values of dimensions in (b) are listed in Table. e

4 4 T. Takahashi Physical Review E., v.6 (000) 8606 The angular distribution of radiation from the quartz of 60 mm long at λ=1.3 mm and the Teflon of 100 mm long at λ=.0 mm. The data are plotted on a logarithmic scale in order to visualize satellite peaks in the angular distribution The dependence of the angular distribution on the length of quartz. The solid, broken, and dotted curves represent the data for quartz of 60, 40, and 0 mm long, respectively. The curves at the right-hand side show the theoretical calculation.

5 Potylitsyn et al. Journal of Physics: Conference Series 36 (010) 0105 6.

$151 mm focal distance DP-1M detector 1 17 mm viewed a range of wavelengths 1.45 Teflon refractive index Experimental scheme and some definitions.$

5 5 Potylitsyn et al. Journal of Physics: Conference Series 36 (010) Mev (γ 1) electron energy 30 ma average beam current 10 8 maximal bunch population 1056 Bunches in a train 1.1 mm longitudinal size (rms) of electrons 4 4 mm transverse sizes of electron beam 4 µs train duration 5 mm impact-parameter 170 mm Diameter of paraboloidal mirror 151 mm focal distance DP-1M detector 1 17 mm viewed a range of wavelengths 1.45 Teflon refractive index Experimental scheme and some definitions. The angular dependence of CChR: the red circles horizontal polarization component, the blue squares vertical polarization component, the solid line theoretical simulation.

6 K. Sakaue, et al., in Proc. of IPAC 16, Busan, Korea (016), TUPOW047.

$5 TOPAS refractive index 48.$

6 6 K. Sakaue, et al., in Proc. of IPAC 16, Busan, Korea (016), TUPOW Mev (γ 10) electron energy quasi-optical detector 0.1 THz viewed a range of frequency 1.5 TOPAS refractive index 48.5 deg Cherenkov angle width 1 mm Prizm target size thickness 1 mm THz intensity with 1 THz as a function of the electron bunch position and electron bunch tilting angle Experimental setup for coherent Cherenkov radiation by using tilted electron bunch.

7 7 Incoherent Cherenkov radiation dw ChR dω dω uur ur E ( r ω) = cr ' vac, (1) 1 uuur uur (, ) R uur (, R uur Evac r ω = εfe H r ω) FH H ( r, ω ) ε P + () ur ur ur ur (, ω) = z (, ω) + x (, ω) sin ( φ) + y (, ω) cos( φ) ( ) R R R R HP r H r H r H r (3) ur ur ur = (, ω) x (, ω) cos ( φ) y (, ω) sin( φ) (4) R R R H r H r H r εcos F = H εcos + ε sin (5) F E = cos cos + ε sin (6)

8 8 ω i r' ε / / ( 1 c L a + H a ε ) ω e r ik ( ) * y y+ kz z ik ( y y+ kz z ) uur R uur e uur e H ( kx, y, z, ω) = k ( kx, y, z, ω) e dy ( kx, y, z, ω) e dy dz (7) c r' E + E 0 a/ a/ H ur E ( n ) { ( ) } ie ( k, y, z, ω) = εβγn, i 1 + ε βγn, γ e e x x x πβc 1+ ε βγ * e 1 x ω ω i z y 1+ ε βγn x βс βсγ ( ) (8) ur E e ( n ) { ( ) } 1 ie ( k, y, z, ω) = εβγn, i 1 + ε βγn, γ e e x x x πβc 1+ ε βγ x ω ω i z y 1+ ε βγn x βс βсγ ( ) (9) r r ω r 1 k = n ε, n= { sin( ) sin ( φ), sin( ) cos ( φ), ε sin ( ) } (10) c ε

9 9 ( 1+ γ sin ) ( ε 1) ( ( )) iπl 1 β ε sin ( ) ( ) sin ( φ) β ε ( ( )) ChR e β cos = ω Ω 4π 1 β ε sin dw d d c e λβ 1 Spectral and angular distribution of the radiation, calculated using the method of polarization currents [D. V. Karlovets, А. P. Potylitsyn, Phys. Lett. A 373, 1988 (009).] ε ( Φ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) iγ cos φ ε sin 1+ γ sin sin φ β + Φ 1 sin cos φ + sin sin φ γ 1 β β ε sin ( ) εcos( ) + ε sin ( ) ε cos + ε sin sin ( ) sin ( φ ) γ Φ1 cos( φ) sin( ) s ( ) 1 ε ( ) 1, ε ( ( )) + ( ) ( ) ( Φ1γsin( ) cos( φ) β +Φ i 1+ γ sin ( ) sin ( φ) β ) + ( sin( φ) ( Φ1sin( ) cos( φ) + iφγ ε sin ( ) 1+ γ sin ( ) sin ( φ) β ) in φ γ ( β β sin )) (11) ( ( ) ( ) ( ) ( ) ) ( ( ) ( ) ( ) ( ) ) aπ iγsin cos φ β 1 γ sin sin φ β πh iγsin cos φ β 1 γ sin sin φ β exp 1 exp γλβ γλβ Φ = ± ( iγ Únyβ + 1+ γ sin ( ) sin ( φ) β ) ( i sin( ) cos( φ) 1 sin ( ) sin ( φ + + ) ) H i sin( ) cos( φ) + 1+ sin ( ) sin ( φ) aπ γ β γ β π γ exp 1 exp γλβ ± iγsin cos φ β + 1+ γ sin sin φ β ( ( ) ( ) ( ) ( ) ) ( β γ β ) γλβ (1)

10 10 V.E. Pafomov, JETP, 33, 1074 (1957) (Russian) dw ChR dω dω = ( ) cos ( ) ( ε 1) ( ( )) ( 1 cos ( )) 1 sin ( ) ( ) iπl ε sin ( ) e β sin π c β β ε ( ( ) ( )) ( ) ( ( ) ( )) ( )( ( )) e λ ε sin εcos 1 β ε sin 1 β + β ε sin iπl ε sin e Spectral and angular distribution of the radiation, calculated using method of images iπl ε sin iπl ε sin λ λ ( ) ( ( ) ( )) ε sin + εcos e ε sin εcos e + e λ ( ) ( ε sin ( ) + εcos( ) )( 1+ β ε sin ( ) ) 1 β β ε sin ( ) ( ) ( ( ) ( )) iπl ε sin iπl ε sin λ λ ( ) ( ) ( ( ) ( )) ε sin + εcos e ε sin εcos e ( ) iπ L λβ ( ( ) ( )) ( β β ε) ( ) ( ( )) 1 cos( ) e ε sin 1 + βcos iπl ε sin iπl ε sin λ λ ( ) ( ( ) ( )) ε sin + εcos e ε sin εcos (13)

11 11 Comparison of both models H= mm a=0 mm L=5 mm γ=10 ε=1.3 λ=1.5 mm φ = 0 o polarization currents Pafomov H= mm a=0 mm L=75 mm γ=10 ε=1.3 λ=1.5 mm φ = 0 o polarization currents Pafomov

12 1 Angular distribution of incoherent ChR r k { ( ) ( ) ( ) ( ( ) ( ))} sin x, cos x sin y, ε 1 cos x cos y π = λ H=50 mm L=5 mm a=0 mm γ=10 ε=1.3 λ=1.5 mm

13 13 H=50 mm L=5 mm a=0.1 mm γ=10 ε=1.3 λ=1.5 mm H=50 mm L=5 mm a=5 mm γ=10 ε=1.3 λ=1.5 mm

14 14 see, Norihiro Sei, Takeshi Sakai, et. al., Physics Letters A, 379 (015) can be approximated as exp 4πa 4πa exp = exp = exp( a ) γλβ H=50 mm L=5 mm γ=10 ε=1.3 λ=1.5 mm 4π a γλβ 4πa 4πa exp = exp = exp γλβ H=50 mm L=5 mm γ=50 ε=1.3 λ=1.5 mm ( a ) Fit exp a polarization currents Fit exp a polarization currents

15 15 Coherent Cherenkov radiation The spectral-angular density of Coherent Cherenkov radiation (CChR) from a bunch with population N: r ( + ( )) CChR dw dwchr = N 1 ( N 1) F k, dωdω dωdω r r Fo rm factor Fk ( ) ( r) exp i ω sin( ) sin( ) x ω sin( ) cos ( ) y ω r = ρ φ + φ + z dr, c c cβ dw ChR dωdω is the spectral-angular density for a single charge. r 1 1 x y z ρ() r = ρ(, x y,) z = exp + +. (*) 3/ ( π) σ σ xσyσ z x σ y σ z

16 16 Form factor a tilted electron bunch with distribution of electrons in the bunch is a Gaussian: π z Vϕ = xsin ( ) sin ( φ) + ysin ( ) cos ( φ) + λ β

17 17 Form factor for tilted electron bunch r r Fk ( ) ( r)exp i ω sin( ) sin( ) x ω sin( ) cos ( ) y ω r = ρ φ + φ + z dr, c c cβ r 1 1 x ycos( ψ) zsin( ψ) ysin( ψ) + zcos( ψ) ρ() r = exp. (**) 3/ + + ( π) σ σ xσ yσ z x σ y σz r 1 ( ) exp k x x ( ky kz )( y z ) ( ky kz )( y z ) cos( ) kykz( y z ) sin ( ) = , (14) Fk σ σ σ σ σ ψ σ σ ψ π 1,, = sin, cos, (15) λ β { kx ky kz} sin( ) ( φ) sin( ) ( φ) π 1,, = sin,cos sin, (16) λ β { kx ky kz} ( x) ( x) ( y)

18 18 Angular distribution of coherent Cherenkov radiation from a tilted bunch passing through a target CChR H=50 mm L=5 mm a=0 mm γ=10 ε=1.3 λ=1.5 mm σx= σy=707 µm σz= 100 µm ψ=0 deg CChR H=50 mm L=5 mm a=0 mm γ=10 ε=1.3 λ=1.5 mm σx= σy=707 µm σz= 100 µm ψ=3.6 deg

19 19 Angular distribution of coherent Cherenkov radiation from a tilted bunch passing through a slit in target CChR H=50 mm L=5 mm a=5 mm γ=10 ε=1.3 λ=1.5 mm σx= σy=707 µm σz= 100 µm ψ=0 deg CChR H=50 mm L=5 mm a=5 mm γ=10 ε=1.3 λ=1.5 mm σx= σy=707 µm σz= 100 µm ψ=3.6 deg

20 3 z ( ) b i i π π Vϕ = sin x 1 ( βγ sin ( )) y λ + + β βγλ x b x b H=50 mm a=0 mm γ=100 ε=1.3 λ=1.5 mm σx= σy=707 µm σz= 100 µm

21 0 Coherent Cherenkov radiation from a tilted bunch The schematic view of generation CChR a tilted «pancake-like» electron bunch

22 1 Coherent Cherenkov radiation from a tilted bunch 1 N 1 N 1 N The schematic view of generation CChR a tilted «pancake-like» electron bunch

23 Coherent Cherenkov radiation from a tilted bunch ψ=-8 ψ=8

24 3 Conclusion 1. The developed model allows to simulate the spectral-angular distribution of ChR generated by short electron bunches for which t = y + y? z σ σ σ σ for any target geometry.. ChR produced by an ultrashort electron bunch with the axes tilted relative to the bunch velocity possesses the strong azimuthal asymmetry. 3. Simulation results show that the maximal ChR yield is placed in the plane coinciding with the bunch axes and confirm the experimental data. 4. The observed effect can be used to produce the intense THz radiation beam concentrated in the narrow angular range in contrast with the conventional ChR where an intensity is distributed along the cone surface with the opening angle ChR

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