def _r_analysis(a0, mstack):
"""Compute phase advance and R-matrices in 2D, 4D or 6D
R0, R = _bk_analysis(A0)
PARAMETERS
A0 A-matrix at the beginning of the lattice
mstack transfer matrix from the entrance of the lattice to the selected
element
OUTPUT
R0 R-matrices at the entrance of the lattice
R Iterator over the results at the selected elements
"""
def get_phase(a22):
"""Return the phase for A standardization"""
return atan2(a22[0, 1], a22[0, 0])
def standardize(aa, slcs):
"""Apply rotation to put A in std form"""
rotmat = numpy.zeros((nv, nv))
for slc in slcs:
rot = -get_phase(aa[slc, slc])
cs = cos(rot)
sn = sin(rot)
rotmat[slc, slc] = numpy.array([[cs, sn], [-sn, cs]])
return numpy.dot(aa, rotmat)
def r_matrices(ai):
# Rk = A * S * Ik * inv(A) * S.T
ais = ai.dot(tt) # Longitudinal swap
invai = solve(ai, ss.T)
ri = numpy.array(
[numpy.dot(ais[:, sl], invai[sl, :]) for sl in slices])
return ri
def propagate(ai):
ri = r_matrices(ai)
phi = numpy.array([get_phase(ai[sl, sl]) for sl in slices])
return ri, ai, phi
nv = a0.shape[0]
dms = nv // 2
slices = [slice(2 * i, 2 * (i + 1)) for i in range(dms)]
ss = jmat(dms)
tt = block_diag(*_Tn[:dms]) # Used instead of ss for longitudinal swap
astd = standardize(a0, slices)
inival = (r_matrices(astd), astd, numpy.zeros((dms, )))
return inival, (propagate(mi.dot(astd)) for mi in mstack)
"""
Coupled or non-coupled 4x4 linear motion
"""
import numpy
from math import sqrt, atan2, pi
from at.lattice import Lattice, check_radiation, uint32_refpts, get_s_pos, \
bool_refpts
from at.tracking import lattice_pass
from at.physics import find_orbit4, find_m44, jmat
from .harmonic_analysis import get_tunes_harmonic
__all__ = ['get_twiss', 'linopt', 'avlinopt', 'get_mcf', 'get_tune']
DDP = 1e-8
_jmt = jmat(1)
# dtype for structured array containing Twiss parameters
TWISS_DTYPE = [('idx', numpy.uint32), ('s_pos', numpy.float64),
('closed_orbit', numpy.float64, (6, )),
('dispersion', numpy.float64, (4, )),
('alpha', numpy.float64, (2, )), ('beta', numpy.float64, (2, )),
('mu', numpy.float64, (2, )), ('m44', numpy.float64, (4, 4))]
# dtype for structured array containing linopt parameters
LINDATA_DTYPE = TWISS_DTYPE + [('A', numpy.float64, (2, 2)),
('B', numpy.float64, (2, 2)),
('C', numpy.float64, (2, 2)),
('gamma', numpy.float64)]
def linopt6(ring,
refpts=None,
dp=None,
orbit=None,
cavpts=None,
twiss_in=None,
get_chrom=False,
get_w=False,
keep_lattice=False,
**kwargs):
"""Perform linear analysis of a fully coupled lattice
elemdata0, beamdata, elemdata = linopt6(lattice)
For circular machines, linopt6 analyses
the 4x4 1-turn transfer matrix if radiation is OFF, or
the 6x6 1-turn transfer matrix if radiation is ON.
For a transfer line, The "twiss_in" intput must contain either:
- a field 'R', as provided by ATLINOPT6, or
- the fields 'beta' and 'alpha', as provided by linopt and linopt6
PARAMETERS
ring lattice description.
refpts=None elements at which data is returned.
KEYWORDS
orbit avoids looking for the closed orbit if is already known
((6,) array)
keep_lattice Assume no lattice change since the previous tracking.
Defaults to False
XYStep=1.0e-8 transverse step for numerical computation
DPStep=1.0E-6 momentum deviation used for computation of
the closed orbit
twiss_in=None Initial conditions for transfer line optics. Record
array, as output by linopt or linopt6.
If present, the attribute 'R' will be used, otherwise
The attributes 'alpha' and 'beta' will be used. All
other attributes are ignored.
cavpts=None Cavity location for off-momentum tuning
OUTPUT
elemdata0 linear optics data at the entrance/end of the ring
beamdata lattice properties
elemdata linear optics at the points refered to by refpts, if
refpts is None an empty elemdata structure is returned.
elemdata is a record array with fields:
R R-matrices (3, 6, 6)
A A-matrix (6, 6)
M Transfer matrix from the entrance of the line (6, 6)
beta [betax, betay] vector
alpha [alphax, alphay] vector
dispersion (4,) dispersion vector
mu [mux, muy], betatron phases
s_pos longitudinal position [m]
closed_orbit (6,) closed orbit vector
W (2,) chromatic amplitude function
All values given at the entrance of each element specified in refpts.
Field values can be obtained with either
elemdata['beta'] or
elemdata.beta
beamdata is a record with fields:
tune Fractional tunes
damping_time Damping times [s]
chromaticity Chromaticities
REFERENCES
[1] Etienne Forest, Phys. Rev. E 58, 2481 – Published 1 August 1998
[2] Andrzej Wolski, Phys. Rev. ST Accel. Beams 9, 024001 –
Published 3 February 2006
[3] Brian W. Montague Report LEP Note 165, CERN, 1979
"""
def get_alphabeta(r):
beta = numpy.array([r[0, 0, 0], r[1, 2, 2]])
alpha = -numpy.array([r[0, 1, 0], r[1, 3, 2]])
return alpha, beta
# noinspection PyShadowingNames
def build_r(ring, dp, orbit, refpts=None, mxx=None, **kwargs):
""""""
if ring.radiation:
mt, ms = find_m66(ring, refpts=refpts, orbit=orbit, **kwargs)
else:
mt, ms = find_m44(ring, dp, refpts=refpts, orbit=orbit, **kwargs)
a0, vps = a_matrix(mt if mxx is None else mxx)
val0, vals = _r_analysis(a0, ms)
return ms, vps, val0, vals
def build_sigma(twin):
"""Build the initial distribution at entrance of the transfer line"""
try:
sigm = numpy.sum(twin.R, axis=0)
except AttributeError:
slices = [slice(2 * i, 2 * (i + 1)) for i in range(4)]
ab = numpy.stack((twin.alpha, twin.beta), axis=1)
sigm = numpy.zeros((4, 4))
for slc, (alpha, beta) in zip(slices, ab):
gamma = (1.0 + alpha * alpha) / beta
sigm[slc, slc] = numpy.array([[beta, -alpha], [-alpha, gamma]])
return sigm
def output6(r123, a, mu, *args):
"""Extract output parameters from Bk matrices"""
alpha, beta = get_alphabeta(r123)
dispersion = r123[2, :4, 4] / r123[2, 4, 4]
return (r123, a, alpha, beta, mu, dispersion) + args
def output4(r12, a, *args):
"""Extract output parameters from Bk matrices"""
alpha, beta = get_alphabeta(r12)
return (r12, a, alpha, beta) + args
def get_disp(ringup,
ringdn,
dpup,
dpdn,
refpts=None,
matpts=None,
keep_lattice=False,
**kwargs):
def off_momentum(rng, dp):
orb0, orbs = _find_orbit(rng,
dp,
refpts=refpts,
keep_lattice=keep_lattice,
**kwargs)
dp = orb0[4] # in 6D, dp comes out of find_orbit6
_, vps, el0, els = build_r(rng,
dp,
orb0,
refpts=matpts,
keep_lattice=True,
**kwargs)
tunes = numpy.mod(numpy.angle(vps) / 2.0 / pi, 1.0)
return dp, tunes, orb0, orbs, el0, els
def chromfunc(ddp, elup, eldn):
aup, bup = get_alphabeta(elup[0])
adn, bdn = get_alphabeta(eldn[0])
db = (bup - bdn) / ddp
mb = (bup + bdn) / 2
da = (aup - adn) / ddp
ma = (aup + adn) / 2
w = numpy.sqrt((da - ma / mb * db)**2 + (db / mb)**2)
return w
dpup, tunesup, o0up, orbup, el0up, elup = off_momentum(ringup, dpup)
dpdn, tunesdn, o0dn, orbdn, el0dn, eldn = off_momentum(ringdn, dpdn)
deltap = dpup - dpdn
chrom = (tunesup - tunesdn) / deltap
disp0 = (o0up - o0dn)[:4] / deltap
disp = ((oup - odn)[:4] / deltap for oup, odn in zip(orbup, orbdn))
w0 = chromfunc(deltap, el0up, el0dn)
w = (chromfunc(deltap, *v) for v in zip(elup, eldn))
return chrom, disp0, disp, w0, w
def unwrap(mu):
"""Remove the phase jumps"""
dmu = numpy.diff(numpy.concatenate((numpy.zeros((1, dms)), mu)),
axis=0)
jumps = dmu < -1.e-3
mu += numpy.cumsum(jumps, axis=0) * 2.0 * numpy.pi
dp_step = kwargs.get('DPStep', DConstant.DPStep)
if twiss_in is None: # Circular machine
mxx = None
else: # Transfer line
if orbit is None:
orbit = numpy.zeros((6, ))
sigma = build_sigma(twiss_in)
mxx = sigma.dot(jmat(sigma.shape[0] // 2))
orb0, orbs = _find_orbit(ring, dp, refpts=refpts, orbit=orbit, **kwargs)
dp = orb0[4]
ms, vps, el0, els = build_r(ring,
dp,
orb0,
refpts=refpts,
mxx=mxx,
keep_lattice=keep_lattice,
**kwargs)
dms = vps.size
length = ring.get_s_pos(len(ring))[0]
tunes = numpy.mod(numpy.angle(vps) / 2.0 / pi, 1.0)
damping_rates = -numpy.log(numpy.absolute(vps))
damping_times = length / clight / damping_rates
spos = get_s_pos(ring, refpts)
if dms >= 3: # 6D processsing
output = output6
dtype = W_DTYPE6
data0 = [*el0, numpy.identity(2 * dms), 0.0, orb0]
datas = [els, iter(ms), iter(spos), iter(orbs)]
if get_chrom or get_w:
f0 = get_rf_frequency(ring, cavpts=cavpts)
df = -dp_step * get_mcf(ring.radiation_off(copy=True)) * f0
rgup = set_rf_frequency(ring,
f0 + 0.5 * df,
cavpts=cavpts,
copy=True)
rgdn = set_rf_frequency(ring,
f0 - 0.5 * df,
cavpts=cavpts,
copy=True)
if get_w:
dtype = W_DTYPE6 + W_DTYPEW
chrom, _, _, w0, ws = get_disp(rgup,
rgdn,
None,
None,
matpts=refpts,
**kwargs)
data0.append(w0)
datas.append(ws)
else:
chrom, _, _, _, _ = get_disp(rgup, rgdn, None, None)
else:
chrom = numpy.NaN
else: # 4D processsing
output = output4
dtype = W_DTYPE4
data0 = [*el0]
datas = [els]
if get_w:
dtype = W_DTYPE4 + W_DTYPEW
chrom, d0, ds, w0, ws = get_disp(ring,
ring,
dp + 0.5 * dp_step,
dp - 0.5 * dp_step,
refpts=refpts,
matpts=refpts,
keep_lattice=True,
**kwargs)
data0 += [d0, numpy.identity(2 * dms), 0.0, orb0, w0]
datas += [ds, iter(ms), iter(spos), iter(orbs), ws]
elif get_chrom:
chrom, d0, ds, w0, ws = get_disp(ring,
ring,
dp + 0.5 * dp_step,
dp - 0.5 * dp_step,
refpts=refpts,
keep_lattice=True,
**kwargs)
data0 += [d0, numpy.identity(2 * dms), 0.0, orb0]
datas += [ds, iter(ms), iter(spos), iter(orbs)]
else:
chrom = numpy.NaN
data0 += [numpy.NaN, numpy.identity(2 * dms), 0.0, orb0]
datas += [repeat(numpy.NaN), iter(ms), iter(spos), iter(orbs)]
elemdata0 = numpy.array(output(*data0), dtype=dtype).view(numpy.recarray)
elemdata = numpy.fromiter((output(*el, *v) for el, *v in zip(*datas)),
dtype,
count=ring.refcount(refpts)).view(numpy.recarray)
beamdata = numpy.array(
(tunes, chrom, damping_times),
dtype=[('tune', numpy.float64, (dms, )),
('chromaticity', numpy.float64, (dms, )),
('damping_time', numpy.float64, (dms, ))]).view(numpy.recarray)
unwrap(elemdata.mu)
return elemdata0, beamdata, elemdata
A collection of functions to compute 4x4 and 6x6 transfer matrices
"""
import numpy
from at.lattice import Lattice, uint32_refpts, get_refpts, check_radiation
from at.lattice.elements import Bend, M66
from at.tracking import lattice_pass, element_pass
from at.physics import find_orbit4, find_orbit6, jmat, symplectify
__all__ = ['find_m44', 'find_m66', 'find_elem_m66', 'gen_m66_elem']
XYDEFSTEP = 6.055454452393343e-006 # Optimal delta?
DPSTEP = 6.055454452393343e-006 # Optimal delta?
_jmt = jmat(2)
def find_m44(ring,
dp=0.0,
refpts=None,
orbit=None,
keep_lattice=False,
**kwargs):
"""find_m44 numerically finds the 4x4 transfer matrix of an accelerator
lattice for a particle with relative momentum deviation DP
IMPORTANT!!! find_m44 assumes constant momentum deviation.
PassMethod used for any element in the lattice SHOULD NOT
1. change the longitudinal momentum dP
(cavities , magnets with radiation, ...)