1. Optics and magnetic field calculation for the Hall D Tagger Guangliang Yang Glasgow University

2. Contents 1. Magnetic field calculated using Opera 3D. 2. Tagger optics calculated using Opera 3D. 3. Tagger optics along the straight line focal plane. 4. Effects of position and direction errors on the straight line focal plane optics. 5. Conclusion.

3. Part 1. Magnetic field calculation. The magnetic field of the Hall D Tagger is calculated by using a finite element software- Opera 3D, version 10.025 Two identical dipoles and one quadupole are included in the same mesh model. More than 2 million elements and 1.5 million nodes have been used in the calculation. The magnetic fields have been checked along various electron trajectories.

4. Magnetic field calculated by using Opera 3D, version 10.025. Mesh used by Tosca for magnetic field calculation.

5. Mid-plane magnetic field histogram calculated by TOSCA. Magnetic field along a line perpendicular to the magnet output edge. TOSCA Magnetic Field Calculation.

6. Magnetic field along electron beam trajectory (1GeV).

7. Magnetic field along electron beam trajectory (8 GeV).

8. Magnetic field along electron beam trajectories between 3.9 and 5.0 GeV.

9. Y-component of stray field at focal plane position. Minimum distance between focal plane detector and EFB

10. Component of stray field normal to y-direction at focal plane position. Minimum distance between focal plane detector and EFB

11. Part 2. Optics calculated using Opera 3D. The electron trajectories of various energies have been evaluated using the calculated magnetic field. By using the calculated electron trajectories, optical properties of the Tagger are determined.

12. Starting position and direction of an electron trajectory. We use (x, y, z) to describe the starting position of an electron trajectory and use α and ψ to determine its direction. (x, y, z) are the co-ordinates of a point in a Cartesian system. The positive z direction is along the main beam direction, the y direction is perpendicular to the mid plane of the tagger, and the positive x direction is opposite to the bending direction. α is the angle between the projected line of the emitted ray on the x-z plane and the z axis, ψ is the angle between the projected line of the emitted ray on the y-z plane and the z axis.

13. Ray bundle used in the calculation By varying x, y, α and ψ, 81 trajectories are defined for each bundle. x=σ x or 0 or -σ x. y=σ y or 0 or -σ y. z=-300 cm (i.e. the radiator position). α =4σ h or 0 or -4σ h. ψ =4σ v or 0 or -4σ v. σ x and σ y are the standard deviations for the main beam in the horizontal or vertical directions. σ h and σ v are the energy degraded electron characteristic angles in the horizontal or vertical directions.

14. Electron trajectories have been calculated using Opera 3 D post processor. The focal plane position is determined by using the calculated electron trajectories and spot sizes. Beam trajectories (1-9 GeV) and the straight line focal plane position Tosca. Different colours indicate different energies

15. Calculated electron trajectories (81 per ray bundle). Beam trajectories calculated from TOSCA in the mid plane for 3 GeV and 8 GeV. Those trajectories having the same direction focus on position 1, and those trajectories having the same starting position focus on position 2. ( Electrons travelling in the direction shown by the top arrow ). Electron trajectory bundles according to their directions at the object position. (3 GeV)(8 GeV) 12 1 2

16. Object Image Sketch showing the two focusing positions Position 1 Position 2 Lens From the TOSCA calculation, the best location for a straight line focal plane is close to position 2 for the lower electron energies. For high electron energies the best location is close to position 1.

17. Beam trajectories calculated by TOSCA in a vertical plane for 3 GeV electrons. Exit edge Focal plane Without quadrupole With quadrupole Rays with different starting points but with a common angle Y position depends on emission angle of bremsstrahlung electrons.

18. The intersections of the beam trajectories with the plane through the focusing point for the central line energy and perpendicular to the beam. TOSCA calculation of the beam spot profile at the focal plane. For 3 GeV electrons and no quadruople. The nearest three intersections are in the same group. The intersections in the same group have the same x and y co-ords at the radiator but a different angle α. They are superimposed together at the precise point to point focus position. different lab y co-ords at radiator Different angles ψ at radiator Different lab x co-ords at radiator. (without Quadrupole)

19. TOSCA calculation of the beam spot profile at the focal plane for 3 GeV electrons and with a quadrupole (81 lines). 9 intersections superimposed together Different lab x co-ords at radiator Different angles ψ at radiator With quadrupole

20. Two identical dipoles Tagger (with the quadrupole adjusted to focus at 3 GeV).

21. Two identical dipoles Tagger (with the quadrupole adjusted to focus at 4.3 GeV).

22. Envelopes of electron beam trajectories as they cross the focal plane– using 81 trajectory ray bundles (without quadrupole).

23. Electron beam trajectories as they cross the focal plane - central ray only.

24. Par 3. Tagger optics along the straight line focal plane. 1.The optical properties have been determined using Tosca ray tracing. 2.The optical properties meet the requirements of GlueX.

25. Straight line focal plane position Main beam Magnet 1 Photon beam Straight thin window flange (parallel to the straight line focal plane determined by TOSCA ray tracing)

26. Comparison of optical properties along the Straight Line focal planes (without and with quadrupole). (quadrupole field optimized for 3 GeV electrons.) Resolution.Half vertical height.

27. Dispersion.Beta. Comparison of optical properties along the Straight Line focal plane (without and with quadrupole). Beta is the angle between an outgoing electron trajectory and the focal plane. (Perpendicular to electron trajectory)

28. Part 5. Effects of positioning errors. The effects of positioning errors on the Tagger optics are simulated by using Opera 3 D. In these calculations, the second magnet is intentionally put in the wrong position. Various positioning errors have been investigated: 1. the second magnet is moved longitudinally +-2 mm along a straight line parallel to the long exit edge of the first magnet. 2. the second magnet is moved right or left 2 mm along a straight line perpendicular to the long exit edge of the first magnet. 3. the second magnet is rotated around the bottom right corner of the second magnet by an angle of 0.1 degree or -0.1degree. It has been found that the Tagger optical properties are insensitive to the positioning errors.

29. Conclusions The optical properties along the straight line focal plane of the two identical magnets Tagger meet the GlueX specifications. The Tagger optical properties are insensitive to the positioning errors investigated.