1. 1 IR with elliptical compensated solenoids in FCC-ee S. Sinyatkin Budker Institute of Nuclear Physics 13 July 2015, CERN

2. 2 Task Solenoid fringe field effect is estimated. Simple optical model of elliptical solenoids is presented. Magnetic field simulation of ordinary and elliptical solenoids is carried out. Effect of the ordinary and elliptical solenoids on the ring optics and beam emittance for version with location of the anti-solenoids between the FF quad and IP is considered.

3. 3 Compensating solenoid is located between the IP and QD0 Transverse half size: - compensated solenoid - Rx = 0.1 m, Ry = 0.025 m - screening solenoid - Rx = 0.2 m, Ry = 0.2 m

4. 4 Representation of elliptical solenoids for linear optics by means of ordinary solenoid and skew quad

5. 5 Linear part to use in linear optics by MAD

6. 6 Representation of elliptical solenoids for linear optics by means of ordinary solenoid and skew quads Elliptical solenoid Ordinary solenoid + Skew quad G skew - strength of skew quad B s ’ –derivation of longitudinal magnetic field

7. 7 Representation of elliptical solenoids for linear optics by means of ordinary solenoid and skew quads Elliptical solenoid with magnetic field B s and length L. Ordinary solenoid: - with magnetic field B s - length L. Thin skew quads: Strength of skew G_skew in = - G_skew out = G_skew L skew ~ 2∙R y, R y < R x G_skew -G_skew Elliptical Solenoid Ordinary Solenoid

8. 8 Parameters of solenoids Main solenoid field: L = 2 m, B = 2 T 2 compensating solenoids, each of L = 1 m, B = - 2 T (B max = - 4 T) Screening solenoid covers the FF quadrupoles Transverse half size: - compensated solenoid - Rx = 0.1 m, Ry = 0.025 m - screening solenoid - Rx = 0.2 m, Ry = 0.2 m Angle between the beams reference trajectory and axis of solenoid ±15 mrad Beams at the compensating solenoid entry (from IP) are displaced horizontally for 1000. sin(±0.015) ≈ ±15 mm

9. 9 Magnetic field simulation of solenoids Y S XIP Rx 2*Ry Y S X IP Transverse half size: - compensated solenoid - Rx = 0.1 m, Ry = 0.025 (0.1-right) m - screened solenoid - Rx = 0.2 m, Ry = 0.2 m Length: - compensated solenoid - L = 1 m - screened solenoid - L = 4.5 m Size of edge field area is ~ min( Rx, Ry )

10. 10 Longitudinal field distribution along trajectory Solenoids: 1 – screen. solenoid (Rx=Ry=0.2 m), comp. Solenoid (Rx=0.1 m, Ry=0.1 m) w/o rotation. 2 – screen. solenoid (Rx=Ry=0.2 m), comp. Solenoid (Rx=0.1 m, Ry=0.1 m) with rotation. 3 – screen. solenoid (Rx=Ry=0.2 m), comp. Solenoid (Rx=0.1 m, Ry=0.025 m) with rotation. 4 – screen. solenoid (Rx=0.8 m, Ry=0.2 m), comp. Solenoid (Rx=0.1 m, Ry=0.025 m) with rotation. 5 – screen. solenoid (Rx=0.3 m, Ry=0.3 m), comp. Solenoid (Rx=0.15 m, Ry=0.024 m) with rotation. Edge field area for elliptical solenoid is small in comparison with ordinary solenoids. Area of fringe field is minimal transverse size of solenoid ~ min(Rx,Ry). Correction of fringe field at QD0 is necessary.

11. 11 Radial field distribution along reference trajectory - Maximal B x for ordinary solenoid. -Minimal B x for elliptical solenoid. -If compensated and screened solenoids are of the same type the integral of B x is zero. - In case of the different type of compensated and screening solenoids the integral is nonzero. Vertical dispersion and vertical orbit is non zero after solenoids.

12. 12 Distribution of skew component Elliptical solenoid creates large skew quadrupole component of magnetic field at the ends. Rotation of solenoids creates small skew quadrupole component at the ends.

13. 13 Optic model of solenoids To insert piecewise elements into 2 m (distance from IP to QD0) the length is decreased by 10 %. Solenoids are presented by thick elements. Skew component is thin element. Radial field is thin element with nonzero length (Lrad).

14. 14 Beam parameters (E=45 GeV) Version012345 Main solenoid B, T L, m 2323 2222 2222 2222 2222 2222 Comp. solenoid L, m B, T Rx, m Ry, m Tilt 0.5 -6 0.1 No Tilt 1 -4 0.1 Tilt 1 -4 0.1 Tilt 1 -4 0.1 0.025 Tilt 1 -4 0.1 0.025 Tilt 1 -4 0.15 0.024 Screen. solenoid B, T Rx, m Ry, m Rot. -2 0.2 Rot. -2 0.2 Rot. -2 0.2 Rot. -2 0.2 Rot. -2 0.8 0.2 Rot. -2 0.3 Betatron tunes: Qx Qy 110.540 No sol.110.535No sol. 87.60887.586 87.587 Betafunction IP: Betx, m0.50 0.57 Bety, mm1.481.21 1.23 Emittance, nm*rad: horizontal0.1060.109 0.102 vertical1.68E-22.32E-132.04E-49.88E-3 V/H Emittance 0.1590.01.88E-30.097

15. 15 Beam parameters (E=175 GeV) Version012345 Main solenoid B, T L, m 2323 2222 2222 2222 2222 2222 Comp. solenoid L, m B, T Rx, m Ry, m Tilt 0.5 -6 0.1 No Tilt 1 -4 0.1 Tilt 1 -4 0.1 Tilt 1 -4 0.1 0.025 Tilt 1 -4 0.1 0.025 Tilt 1 -4 0.15 0.024 Screen. solenoid B, T Rx, m Ry, m Rot. -2 0.2 Rot. -2 0.2 Rot. -2 0.2 Rot. -2 0.2 Rot. -2 0.8 0.2 Rot. -2 0.3 Betatron tunes: Qx Qy 110.540 110.539 87.57387.571 87.57087.57187.569 Betafunction IP: Betx, m0.50 0.510.500.52 Bety, mm1.041.02 1.031.021.03 Emittance, nm*rad: horizontal1.661.661.66 1.521.631.49 vertical0.42E-30.45E-130.42E-50.180.030.25 V/H Emittance 2.5E-40.02.53E-60.1201.98E-20.169

16. 16 Representation field Solenoids edge 1 2 Bx = 0.03 T Bx = -0.09 T Simple presentation of solenoids with single thin edge vertical kick. Strength vertical kicks are placed at large beta functions in comparison with distributed presentation. Emittance for this case is larger. Distributed presentation of solenoids with several thin edge vertical kicks.

17. 17 Twiss functions Single edge vertical kick Distributed edge vertical kicks

18. 18 Tilts of eigen modes Angle between axis of ordinary solenoids and reference trajectory is zero. Angle between axis of ordinary solenoids and reference trajectory is 15 mrad. There is small betatron coupling. Angle between axis of solenoids (ordinary – screened solenoid; elliptical – compensated solenoid) and reference trajectory is non zero. The field of main solenoid is not compensated by an elliptical solenoid. deg

19. 19 Summary Elliptical solenoid can be represented by ordinary solenoid and skew quads. To reduce vertical emittance field of compensated solenoid is reduced to - 2 T and its length is increased to 1 m. Integral of the radial field created by elliptical solenoid is non zero. In case of an elliptical solenoids for compensated and screened ones the integral of radial field is reduced. Betatron coupling is not suppressed by elliptical compensated solenoid. Splitting allows to take into account a fringe field of solenoids more precisely. For design luminosity the length of compensated solenoid should be 1 m.