LINOPT performs linear analysis of the COUPLED lattices
Notation is the same as in reference [3]
LinData = LINOPT(RING,DP,REFPTS) is a MATLAB structure array with fields
ElemIndex - ordinal position in the RING
SPos - longitudinal position [m]
ClosedOrbit - closed orbit column vector with
components x, px, y, py (momentums, NOT angles)
Dispersion - dispersion orbit position vector with
components eta_x, eta_prime_x, eta_y, eta_prime_y
calculated with respect to the closed orbit with
momentum deviation DP
M44 - 4x4 transfer matrix M from the beginning of RING
to the entrance of the element for specified DP [2]
A - 2x2 matrix A in [3]
B - 2x2 matrix B in [3]
C - 2x2 matrix C in [3]
gamma - gamma parameter of the transformation to eigenmodes
mu - [ mux, muy] horizontal and vertical betatron phase
beta - [betax, betay] vector
All values are specified at the entrance of each element specified in REFPTS.
REFPTS is an array of increasing indexes that select elements
from the range 1 to length(LINE)+1.
See further explanation of REFPTS in the 'help' for FINDSPOS
[LinData,NU] = LINOPT() returns a vector of linear tunes
[nu_u , nu_v] for two normal modes of linear motion [1]
[LinData,NU, KSI] = LINOPT() returns a vector of chromaticities ksi = d(nu)/(dP/P)
[ksi_u , ksi_v] - derivatives of [nu_u , nu_v]
See also FINDSPOS TWISSRING TUNECHROM
[1] D.Edwars,L.Teng IEEE Trans.Nucl.Sci. NS-20, No.3, p.885-888, 1973
[2] E.Courant, H.Snyder
[3] D.Sagan, D.Rubin Phys.Rev.Spec.Top.-Accelerators and beams, vol.2 (1999)0001 function [LinData, varargout] = linopt(RING,DP,varargin); 0002 %LINOPT performs linear analysis of the COUPLED lattices 0003 % Notation is the same as in reference [3] 0004 % 0005 % 0006 % LinData = LINOPT(RING,DP,REFPTS) is a MATLAB structure array with fields 0007 % 0008 % ElemIndex - ordinal position in the RING 0009 % SPos - longitudinal position [m] 0010 % ClosedOrbit - closed orbit column vector with 0011 % components x, px, y, py (momentums, NOT angles) 0012 % Dispersion - dispersion orbit position vector with 0013 % components eta_x, eta_prime_x, eta_y, eta_prime_y 0014 % calculated with respect to the closed orbit with 0015 % momentum deviation DP 0016 % M44 - 4x4 transfer matrix M from the beginning of RING 0017 % to the entrance of the element for specified DP [2] 0018 % A - 2x2 matrix A in [3] 0019 % B - 2x2 matrix B in [3] 0020 % C - 2x2 matrix C in [3] 0021 % gamma - gamma parameter of the transformation to eigenmodes 0022 % mu - [ mux, muy] horizontal and vertical betatron phase 0023 % beta - [betax, betay] vector 0024 % 0025 % All values are specified at the entrance of each element specified in REFPTS. 0026 % REFPTS is an array of increasing indexes that select elements 0027 % from the range 1 to length(LINE)+1. 0028 % See further explanation of REFPTS in the 'help' for FINDSPOS 0029 % 0030 % [LinData,NU] = LINOPT() returns a vector of linear tunes 0031 % [nu_u , nu_v] for two normal modes of linear motion [1] 0032 % 0033 % [LinData,NU, KSI] = LINOPT() returns a vector of chromaticities ksi = d(nu)/(dP/P) 0034 % [ksi_u , ksi_v] - derivatives of [nu_u , nu_v] 0035 % 0036 % See also FINDSPOS TWISSRING TUNECHROM 0037 % 0038 % [1] D.Edwars,L.Teng IEEE Trans.Nucl.Sci. NS-20, No.3, p.885-888, 1973 0039 % [2] E.Courant, H.Snyder 0040 % [3] D.Sagan, D.Rubin Phys.Rev.Spec.Top.-Accelerators and beams, vol.2 (1999) 0041 0042 0043 0044 NE=length(RING); 0045 if(nargin==2) 0046 REFPTS= 1; 0047 else 0048 REFPTS=varargin{1}; 0049 end 0050 0051 NR=length(REFPTS); 0052 0053 0054 spos = findspos(RING,REFPTS); 0055 [M44, MS, orb] = findm44(RING,DP,REFPTS); 0056 0057 LinData = struct('ElemIndex',num2cell(REFPTS),'SPos',num2cell(spos),... 0058 'ClosedOrbit',num2cell(orb,1),'M44',squeeze(num2cell(MS,[1 2]))'); 0059 0060 % Calculate A,B,C, gamma at the first element 0061 M =M44(1:2,1:2); 0062 N =M44(3:4,3:4); 0063 m =M44(1:2,3:4); 0064 n =M44(3:4,1:2); 0065 0066 % 2-by-2 symplectic matrix 0067 S = [0 1; -1 0]; 0068 H = m + S*n'*S'; 0069 t = trace(M-N); 0070 0071 g = sqrt(1 + sqrt(t*t/(t*t+4*det(H))))/sqrt(2); 0072 G = diag([g g]); 0073 C = -H*sign(t)/(g*sqrt(t*t+4*det(H))); 0074 A = G*G*M - G*(m*S*C'*S' + C*n) + C*N*S*C'*S'; 0075 B = G*G*N + G*(S*C'*S'*m + n*C) + S*C'*S'*M*C; 0076 0077 0078 0079 if REFPTS(1)==1 & NR>1 0080 START = 2; 0081 LinData(1).A=A; 0082 LinData(1).B=B; 0083 LinData(1).C=C; 0084 LinData(1).gamma=g; 0085 LinData(1).beta(1) = A(1,2)/sin(acos(trace(A/2))); 0086 LinData(1).beta(2) = B(1,2)/sin(acos(trace(B/2))); 0087 else 0088 START = 1; 0089 end 0090 0091 0092 0093 0094 0095 0096 % find matrixes in all elements indexed by REFPTS 0097 for i=START:NR; 0098 M12 =LinData(i).M44(1:2,1:2); 0099 N12 =LinData(i).M44(3:4,3:4); 0100 m12 =LinData(i).M44(1:2,3:4); 0101 n12 =LinData(i).M44(3:4,1:2); 0102 0103 g2 = sqrt(det(n12*C+G*N12)); 0104 E12 = (G*M12-m12*S*C'*S')/g2; 0105 F12 = (n12*C+G*N12)/g2; 0106 0107 LinData(i).gamma=g2; 0108 LinData(i).C=(M12*C+G*m12)*S*F12'*S'; 0109 LinData(i).A=E12*A*S*E12'*S'; 0110 LinData(i).B=F12*B*S*F12'*S'; 0111 LinData(i).beta(1) = LinData(i).A(1,2)/sin(acos(trace(A/2))); 0112 LinData(i).beta(2) = LinData(i).B(1,2)/sin(acos(trace(B/2))); 0113 0114 end 0115 0116 0117 0118 if nargout > 1 0119 cos_mu_x = trace(A)/2; 0120 cos_mu_y = trace(B)/2; 0121 varargout{1} = acos([cos_mu_x cos_mu_y])/2/pi; 0122 end 0123 0124 if nargout == 3 0125 global NUMDIFPARAMS 0126 0127 if isfield(NUMDIFPARAMS,'DPStep') 0128 dDP = NUMDIFPARAMS.DPStep'; 0129 else 0130 dDP = 1e-8; 0131 end 0132 % Calculate tunes for DP+dDP 0133 [LD, TUNES] = linopt(RING,DP+dDP,1); 0134 varargout{2} = (TUNES - varargout{1})/dDP; 0135 end