Appendix A Quadrupole Doublets, Triplets & Lattices
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1 Appendix A Quadrupole Doublets, Triplets & Lattices George H. Gillespie G. H. Gillespie Associates, Inc. P. O. Box 2961 Del Mar, California 92014, U.S.A. Presented at Sandia National Laboratory (SNL) Albuquerque, New Mexico 18 September 2008 SNL 18 September 2008 Overview of Particle Beam Optics A-1 G. H. Gillespie Associates, Inc.
2 Presentation Outline Overview of Particle Beam Optics Appendix A: Doublets, Triplets and Lattices A.1 Quadrupole Doublet A.2 Quadrupole Triplet A.3 FODO Lattice A.4 Summary SNL 18 September 2008 Overview of Particle Beam Optics A-2 G. H. Gillespie Associates, Inc.
3 A.1. Quadrupole Doublet Two opposite polarity quadrupoles separated by a drift distance L d Antisymmetric doublet: - Quadrupole gradients (G i ) equal in magnitude: G 1 G and G 1 -G - Quad lengths equal: L 1 L 2 L q - Focusing in both planes possible, for certain values of G, L q, L d Antisymmetric Quadrupole Doublet + G - G Non-Antisymmetric Doublets Offer More Flexibility SNL 18 September 2008 Overview of Particle Beam Optics A-3 G. H. Gillespie Associates, Inc.
4 Antisymmetric Quadrupole Doublet Thin lens approximation, the R-matrix for the antisymmetric doublet: - For a quadrupole: f y -f x R xx 1 0-1/f 1 x 1 L d /f 1 x 1+L /f L d x d 2 -L d/f x 1 - L d/f x R yy /f 1 x 1 L d 1 0-1/f 1 x 1-L /f L d x d 2 -L /f 1 +L /f d x d x Thin lens approximation: both planes always focusing f f x 2 / L d > 0 magnifications in two planes different SNL 18 September 2008 Overview of Particle Beam Optics A-4 G. H. Gillespie Associates, Inc.
5 Antisymmetric Quadrupole Doublet Thick lens results R xx cos(kl q) sin(kl q )/k 1 L cosh(kl q) sinh(kl q)/k -k sin(kl q) cos(kl q) d k sinh(kl q) cosh(kl q) M x L eff -1/f M y R yy cosh(kl q) sinh(kl q)/k k sinh(kl q) cosh(kl q) 1 L d cos(kl q) sin(kl q)/k -k sin(kl q) cos(kl q) M y L eff -1/f M x Where the four elements 1/f, M x, M y, L eff are given by: 1/f k[sin(kl q )cosh(kl q ) - cos(kl q )sinh(kl q ) + (kl d )sin(kl q )sinh(kl q )] L eff [sin(kl q )cosh(kl q ) + cos(kl q )sinh(kl q ) + (kl d )cos(kl q )cosh(kl q )]/k M x [cos(kl q )cosh(kl q ) + sin(kl q )sinh(kl q ) + (kl d )cos(kl q )sinh(kl q )] M y [cos(kl q )cosh(kl q ) - sin(kl q )sinh(kl q ) - (kl d )sin(kl q )cosh(kl q )] Apparently the effective focal length f can have either sign: Example kl q π/2, 1/f π[cosh(π/2) + (π/2)(l d /L q )sinh(π/2)]/(2l q ) > 0 Example kl q π, 1/f - π[sinh(π)]/l q < 0 SNL 18 September 2008 Overview of Particle Beam Optics A-5 G. H. Gillespie Associates, Inc.
6 Antisymmetric Quadrupole Doublet + G - G Doublet & Singlet for same initial ray: (q i ) (0.005,0,0.005,0,0,0) T + G SNL 18 September 2008 Overview of Particle Beam Optics A-6 G. H. Gillespie Associates, Inc.
7 Antisymmetric Quadrupole Doublet Condition for positive focal length (f>0) - Equivalently R 12-1/f < 0 or R 21 -k[sin(kl q )cosh(kl q ) - cos(kl q )sinh(kl q ) + (kl d )sin(kl q )sinh(kl q )] < 0 - Since k K 1 1/2 > 0 this condition is - [sin(kl q )cosh(kl q ) - cos(kl q )sinh(kl q ) + (kl d )sin(kl q )sinh(kl q )] < 0 or - (kl d )sin(kl q )tanh(kl q ) < [sin(kl q ) - cos(kl q )tanh(kl q )] - Evidently interested in the case where 0 < kl q < π sin(kl q ) > 0 so (kl d ) > [cos(kl q )/sin(kl q )] - [cosh(kl q )/sinh(kl q )] [cotan(kl q ) - cotanh(kl q )] Effective focal length f positive for "small" kl q, or in terms of R 21 : Thin lens limit (L d >>L q ): R 21 -k[(kl q ) 3 (1 + L d /L q )] < 0 R 21 -k[(kl q ) 3 (L d /L q )] -(k 2 L q ) 2 L d -L d /f x 2 SNL 18 September 2008 Overview of Particle Beam Optics A-7 G. H. Gillespie Associates, Inc.
8 Antisymmetric Quadrupole Doublet + 2G - 2G Two Doublets for same initial ray: (q i ) (0.005,0,0.005,0,0,0) T + G 3 - G SNL 18 September 2008 Overview of Particle Beam Optics A-8 G. H. Gillespie Associates, Inc.
9 A.2 Quadrupole Triplet Three quadrupoles separated by a drift distances, 2 outer quads have same polarity, inner quad opposite polarity to outer quads Symmetric triplet: - Outer quad gradients (G i ) equal: G 1 G and G 3 G - Outer quad lengths (L i ) also equal: L 1 L 3 L q - Two plane focusing possible, for certain values of G, L q, -G c, L q c, L d - Location of principal planes nearly independent of strength Symmetric Quadrupole Triplet + G - G c + G c SNL 18 September 2008 Overview of Particle Beam Optics A-9 G. H. Gillespie Associates, Inc.
10 Example Symmetric Quadrupole Triplet + G - G + G c Triplet & Doublet for same initial ray: (q i ) (0.005,0,0.005,0,0,0) T + G - G SNL 18 September 2008 Overview of Particle Beam Optics A-10 G. H. Gillespie Associates, Inc.
11 A.3. FODO Lattice Periodic array of Antisymmetric Doublets with equal spacing + G - G + G - G Construct FODO lattice out of cells, each cell comprised of an Antisymmetric Doublet with two adjacent Drifts of length L d /2 SNL 18 September 2008 Overview of Particle Beam Optics A-11 G. H. Gillespie Associates, Inc.
12 One Cell of a FODO Lattice Thin lens approximation, the R-matrix for the one FODO cell: R yy 1 L d/2 1-L /f L d x d 2 -L /f 1 +L /f d x d x 1 L d/2 R xx 1-L d/f x -L d/(2f x ) -L d/f x 2 1 L d/ L -L /(2f ) 1+L /f L d x d 2 -L d/f x 1 - L d/f x d d x L /f -L /(2f ) d x d x 1 L d/ L d/f x -L d/(2f x ) 2L d-l d/(2f x ) 2 2 -L d/f x 2 1-L d/f x -L d/(2f x ) Focusing in both planes, but is the motion stable over repeated cells? - Trace of submatrices: Tr[R xx ] Tr[R yy ] 2[1-L d 2 /(2f x 2 )] - Stability condition ( (1/2)Tr[R] 1): [1-L d 2 /(2f x 2 )] 1 f x L d /2 SNL 18 September 2008 Overview of Particle Beam Optics A-12 G. H. Gillespie Associates, Inc.
13 One Cell of a FODO Lattice With thick quadrupole lenses, the R-matrix for the one FODO cell: R xx 1 L /2 d M x L eff -1/f M y 1 L /2 d M x- L d/(2f) -1/f 2 eff d x y d L -L /(4f) + (M + M )L /2 M - L /(2f) y d R yy has the same form, but with the interchange of M x M y Stability condition ( (1/2)Tr[R] 1) yields: (1/2)Tr[R] (1/2) M x + M y + (L d /f) cos(kl q )cosh(kl q ) + (kl d )[cos(kl q )sinh(kl q ) - sin(kl q )sinh(kl q )] + (1/2)(kL d ) 2 sin(kl q )sinh(kl q ) < 1 SNL 18 September 2008 Overview of Particle Beam Optics A-13 G. H. Gillespie Associates, Inc.
14 Stability Condition for Another Type of "Lattice" Consider four 90 o S-Bends, separated by drifts of length ds The R xx submatrix for each (idealized) S-Bend becomes: R xx cos(hs) sin(hs)/h -hsin(hs) cos(hs) 0 ρ -1/ρ 0 Construct a cell of one S-Bend with drifts of ds/2 on each side: R xx 1 ds/2 0 ρ -1/ρ 0 1 ds/2 2 -ds/(2ρ) ρ -L/(4ρ) -1/ρ -ds/(2 ρ) Stability condition ( (1/2)Tr[R] 1) yields: (1/2)Tr[R] (1/2) -ds/(2ρ) -ds/(2ρ) ds/(2ρ) < 1 Stability condition for a simple ring of 4 90 o S-Bends with Reference Trajectory radius ρ, each separated by distance ds: ds < 2ρ SNL 18 September 2008 Overview of Particle Beam Optics A-14 G. H. Gillespie Associates, Inc.
15 Russian Quadruplet Variant of a FODO cell or Two non-antisymmetric doublets with a FODO polarity +G 1 - G 2 +G 2 - G 1 Apparently offers some advantages in reducing aberrations SNL 18 September 2008 Overview of Particle Beam Optics A-15 G. H. Gillespie Associates, Inc.
16 A.4. Summary Overview of Particle Beam Optics Appendix A: Doublets, Triplets and Lattices A.1 Quadrupole Doublet Useful building block A.2 Quadrupole Triplet A.3 FODO Lattice, simple synchrotron lattice Stability condition SNL 18 September 2008 Overview of Particle Beam Optics A-16 G. H. Gillespie Associates, Inc.
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