0 0 1 z (cm) Bz (kG) 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1 50 100 150 200 250 300 350 400 450 5 0 B z (kG) !" !" #" FIG. 25: Design orbit for a 5 GeV positron through the prop osed ILC wiggler. (Upper) Coordinate x (mm) along the length of the wiggler z (m). (Lower) Design orbit in the phase space defined by coordinates x (mm) and p$_{x}$/p $^{0}$. FIG. 23: Fit to the proposed ILC wiggler longitudinal field versus z , where x = 0 . 4 cm, y = 0 . 2 cm. The solid line is com- puted using data on the surface of an elliptical cylinder with x$_{max}$ = 4 . 4 cm, y$_{max}$ = 2 . 4 cm, using the polynomial series for B obtained from (11) or (12). Dots represent numerical data provided by OPERA-3d. FIG. 24: Difference (Gauss) between the vertical field B$_{y}$ of the proposed ILC wiggler and its fitted value across the midplane y = 0. Peak field is 16.7 kG. Lie factorization of the transfer map. Each generator of the symplectic transfer map M is computed in variables represen ting deviations from the design orbit. See the Appendix for a brief discussion of Lie methods. The design orbit is displayed in Fig. 25, and Table I lists its initial and final conditions. Table II displays the matrix R$_{2}$ that describes the linear part of the transfer map in (62). Phase-space coordinates are arranged in the order ( x, p$_{x}$, y, p$_{y}$, τ, p$_{τ}$ ). The first few Lie generators f$_{m}$ of the nonlinear part of the transfer map are listed in Table III. Surface methods provide a reliable and numerically ro- bust technique for extracting transfer maps from numer- ical field data. By benchmarking them against a nu- The on-axis gradients computed for the ILC wiggler were then used in the code MaryLie/IMPACT to inte- grate, simultaneously, i) equations for the design orbit of a 5 GeV positron through the wiggler, ii) equations for the matrix elements of the linear part of the transfer map through the wiggler, and iii) equations for the coefficients of the generating polynomials f$_{3}$ , f$_{4}$ ,... appearing in the B. ILC Damping Ring Wiggler Design Orbit and Associated Transfer Map VII. CONCLUDING SUMMARY 16 0 z (m) ’ref_orbit_’ 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 (mm) ’cesrref’ 2:3 x (mm) - 0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.008 0.006 0.004 0.002 0 -0.00 2 -0.00 4 -0.00 6 -0.00 8 p x  / p 0