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r_matrix.py
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/
r_matrix.py
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__author__ = 'Sergey Tomin'
import logging
from ocelot.common.globals import m_e_GeV, speed_of_light
from ocelot.cpbd.elements import *
logger = logging.getLogger(__name__)
def rot_mtx(angle):
cs = np.cos(angle)
sn = np.sin(angle)
return np.array([[cs, 0., sn, 0., 0., 0.],
[0., cs, 0., sn, 0., 0.],
[-sn, 0., cs, 0., 0., 0.],
[0., -sn, 0., cs, 0., 0.],
[0., 0., 0., 0., 1., 0.],
[0., 0., 0., 0., 0., 1.]])
def uni_matrix(z, k1, hx, sum_tilts=0., energy=0.):
"""
universal matrix. The function creates R-matrix from given parameters.
r = element.l/element.angle
+K - focusing lens, -K - defoc
:param z: element length [m]
:param k1: quadrupole strength [1/m**2]
:param hx: the curvature (1/r) of the element [1/m]
:param sum_tilts: rotation relative to longitudinal axis [rad]
:param energy: the beam energy [GeV]
:return: R-matrix [6, 6]
"""
gamma = energy/m_e_GeV
kx2 = (k1 + hx*hx)
ky2 = -k1
kx = np.sqrt(kx2 + 0.j)
ky = np.sqrt(ky2 + 0.j)
cx = np.cos(z*kx).real
cy = np.cos(z*ky).real
sy = (np.sin(ky*z)/ky).real if ky != 0 else z
igamma2 = 0.
if gamma != 0:
igamma2 = 1./(gamma*gamma)
beta = np.sqrt(1. - igamma2)
if kx != 0:
sx = (np.sin(kx*z)/kx).real
dx = hx/kx2*(1. - cx)
r56 = hx*hx*(z - sx)/kx2/beta**2
else:
sx = z
dx = z*z*hx/2.
r56 = hx*hx*z**3/6./beta**2
r56 -= z/(beta*beta)*igamma2
u_matrix = np.array([[cx, sx, 0., 0., 0., dx/beta],
[-kx2*sx, cx, 0., 0., 0., sx*hx/beta],
[0., 0., cy, sy, 0., 0.],
[0., 0., -ky2*sy, cy, 0., 0.],
[hx*sx/beta, dx/beta, 0., 0., 1., r56],
[0., 0., 0., 0., 0., 1.]])
if sum_tilts != 0:
u_matrix = np.dot(np.dot(rot_mtx(-sum_tilts), u_matrix), rot_mtx(sum_tilts))
return u_matrix
def create_r_matrix(element):
k1 = element.k1
if element.l == 0:
hx = 0.
else:
hx = element.angle / element.l
r_z_e = lambda z, energy: uni_matrix(z, k1, hx=hx, sum_tilts=0, energy=energy)
if element.__class__ == Edge:
sec_e = 1. / np.cos(element.edge)
phi = element.fint * element.h * element.gap * sec_e * (1. + np.sin(element.edge) ** 2)
#phi = element.fint * element.h * element.gap * sec_e * (1. + np.sin(2*element.edge) )
r = np.eye(6)
r[1, 0] = element.h * np.tan(element.edge)
r[3, 2] = -element.h * np.tan(element.edge - phi)
r_z_e = lambda z, energy: r
if element.__class__ in [Hcor, Vcor]:
r_z_e = lambda z, energy: uni_matrix(z, 0, hx=0, sum_tilts=0, energy=energy)
elif element.__class__ == Undulator:
"""
in OCELOT coordinates:
R56 = - Lu/(gamma**2 * beta**2) * (1 + 0.5 * K**2 * beta**2)
S.Tomin, Varenna, 2017.
"""
def undulator_r_z(z, lperiod, Kx, Ky, energy):
gamma = energy / m_e_GeV
r = np.eye(6)
r[0, 1] = z
if gamma != 0 and lperiod != 0 and Kx != 0:
beta = 1 / np.sqrt(1.0 - 1.0 / (gamma * gamma))
omega_x = np.sqrt(2.0) * np.pi * Kx / (lperiod * gamma * beta)
omega_y = np.sqrt(2.0) * np.pi * Ky / (lperiod * gamma * beta)
r[2, 2] = np.cos(omega_x * z)
r[2, 3] = np.sin(omega_x * z) / omega_x
r[3, 2] = -np.sin(omega_x * z) * omega_x
r[3, 3] = np.cos(omega_x * z)
r[4, 5] = - z / (gamma * beta) ** 2 * (1 + 0.5 * (Kx * beta) ** 2)
else:
r[2, 3] = z
return r
r_z_e = lambda z, energy: undulator_r_z(z, lperiod=element.lperiod, Kx=element.Kx, Ky=element.Ky, energy=energy)
# b_z = lambda z, energy: dot((eye(6) - R_z(z, energy)), array([dx, 0., dy, 0., 0., 0.]))
elif element.__class__ == Cavity:
def cavity_R_z(z, V, E, freq, phi=0.):
"""
:param z: length
:param de: delta E
:param f: frequency
:param E: initial energy
:return: matrix
"""
phi = phi * np.pi / 180.
de = V * np.cos(phi)
# pure pi-standing-wave case
eta = 1
# gamma = (E + 0.5 * de) / m_e_GeV
Ei = E / m_e_GeV
Ef = (E + de) / m_e_GeV
Ep = (Ef - Ei) / z # energy derivative
if Ei == 0:
logger.error("CAVITY: Initial energy is 0, check ParticleArray.E or Twiss.E OR cavity.v must be 0")
cos_phi = np.cos(phi)
alpha = np.sqrt(eta / 8.) / cos_phi * np.log(Ef / Ei)
sin_alpha = np.sin(alpha)
cos_alpha = np.cos(alpha)
r11 = (cos_alpha - np.sqrt(2. / eta) * cos_phi * sin_alpha)
if abs(Ep) > 1e-10:
r12 = np.sqrt(8. / eta) * Ei / Ep * cos_phi * sin_alpha
else:
r12 = z
r21 = -Ep / Ef * (cos_phi / np.sqrt(2. * eta) + np.sqrt(eta / 8.) / cos_phi) * sin_alpha
r22 = Ei / Ef * (cos_alpha + np.sqrt(2. / eta) * cos_phi * sin_alpha)
r56 = 0.
beta0 = 1
beta1 = 1
k = 2. * np.pi * freq / speed_of_light
r55_cor = 0.
if V != 0 and E != 0:
gamma2 = Ei * Ei
beta0 = np.sqrt(1. - 1 / gamma2)
gamma2 = Ef * Ef
beta1 = np.sqrt(1. - 1 / gamma2)
#r56 = (beta0 / beta1 - 1) * Ei / (Ef - Ei) * z
r56 = - z/(Ef * Ef * Ei * beta1) * (Ef + Ei)/(beta1 + beta0)
g0 = Ei
g1 = Ef
r55_cor = k * z * beta0 * V / m_e_GeV * np.sin(phi) * (g0 * g1 * (beta0 * beta1 - 1) + 1) / (
beta1 * g1 * (g0 - g1) ** 2)
r66 = Ei/Ef*beta0/beta1
r65 = k*np.sin(phi)*V/(Ef*beta1*m_e_GeV)
cav_matrix = np.array([[r11, r12, 0., 0., 0., 0.],
[r21, r22, 0., 0., 0., 0.],
[0., 0., r11, r12, 0., 0.],
[0., 0., r21, r22, 0., 0.],
[0., 0., 0., 0., 1. + r55_cor, r56],
[0., 0., 0., 0., r65, r66]]).real
if element.coupler_kick:
#element.vxx_up = 1.0003 - 0.8132j
#element.vxy_up = (3.4075 - 0.41223j)
m21 = (element.vxx_up * V * np.exp(1j*phi)).real*1e-3 /E
m43 = - m21
m23 = (element.vxy_up* V * np.exp(1j*phi)).real*1e-3 /E
coupl_kick_up = np.array([[1, 0., 0., 0., 0., 0.],
[m21, 1, m23, 0., 0., 0.],
[0., 0., 1, 0., 0., 0.],
[m23, 0., m43, 1, 0., 0.],
[0., 0., 0., 0., 1., 0.],
[0., 0., 0., 0., 0., 1]]).real
#vxx = ((-4.9278 - 2.2112j) * V * np.exp(1j*phi)).real*1e-3 /(E + de)
#vyy = - vxx
#vxy = ((2.9224 - 0.027228j) * V * np.exp(1j*phi)).real *1e-3 /(E + de)
#element.vxx_down = (-4.9278 - 2.2112j)
#element.vxy_down = (2.9224 - 0.027228j)
m21 = (element.vxx_down * V * np.exp(1j*phi)).real*1e-3 /(E + de)
m43 = - m21
m23 = (element.vxy_down* V * np.exp(1j*phi)).real*1e-3 /(E + de)
coupl_kick_down = np.array([[1, 0., 0., 0., 0., 0.],
[m21, 1, m23, 0., 0., 0.],
[0., 0., 1, 0., 0., 0.],
[m23, 0., m43, 1, 0., 0.],
[0., 0., 0., 0., 1., 0.],
[0., 0., 0., 0., 0., 1]]).real
return np.dot(np.dot(coupl_kick_down, cav_matrix), coupl_kick_up)
return cav_matrix
if element.v == 0.:
r_z_e = lambda z, energy: uni_matrix(z, 0., hx=0., sum_tilts=element.dtilt + element.tilt, energy=energy)
else:
r_z_e = lambda z, energy: cavity_R_z(z, V=element.v * z / element.l, E=energy, freq=element.freq,
phi=element.phi)
elif element.__class__ == TWCavity:
def tw_cavity_R_z(z, V, E, freq, phi=0.):
"""
:param z: length
:param de: delta E
:param f: frequency
:param E: initial energy
:return: matrix
"""
phi = phi * np.pi / 180.
de = V * np.cos(phi)
r12 = z * E / de * np.log(1. + de / E) if de != 0 else z
r22 = E / (E + de)
r65 = V * np.sin(phi) / (E + de) * (2 * np.pi / (speed_of_light / freq)) if freq != 0 else 0
r66 = r22
cav_matrix = np.array([[1, r12, 0., 0., 0., 0.],
[0, r22, 0., 0., 0., 0.],
[0., 0., 1, r12, 0., 0.],
[0., 0., 0, r22, 0., 0.],
[0., 0., 0., 0., 1., 0],
[0., 0., 0., 0., r65, r66]]).real
return cav_matrix
def f_entrance(z, V, E, phi=0.):
phi = phi * np.pi / 180.
de = V * np.cos(phi)
r = np.eye(6)
r[1, 0] = -de / z / 2. / E
r[3, 2] = r[1, 0]
return r
def f_exit( z, V, E, phi=0.):
phi = phi * np.pi / 180.
de = V * np.cos(phi)
r = np.eye(6)
r[1, 0] = +de / z / 2. / (E + de)
r[3, 2] = r[1, 0]
return r
def cav(z, V, E, freq, phi):
R_z = np.dot(tw_cavity_R_z(z, V, E, freq, phi), f_entrance(z, V, E, phi))
R = np.dot(f_exit(z, V, E, phi), R_z)
return R
if element.v == 0.:
r_z_e = lambda z, energy: uni_matrix(z, 0., hx=0., sum_tilts=element.dtilt + element.tilt, energy=energy)
else:
r_z_e = lambda z, energy: cav(z, V=element.v * z / element.l, E=energy, freq=element.freq,
phi=element.phi)
elif element.__class__ == Solenoid:
def sol(l, k, energy):
"""
K.Brown, A.Chao.
:param l: efective length of solenoid
:param k: B0/(2*Brho), B0 is field inside the solenoid, Brho is momentum of central trajectory
:return: matrix
"""
gamma = energy / m_e_GeV
c = np.cos(l * k)
s = np.sin(l * k)
if k == 0:
s_k = l
else:
s_k = s / k
r56 = 0.
if gamma != 0:
gamma2 = gamma*gamma
beta = np.sqrt(1. - 1./gamma2)
r56 -= l/(beta*beta*gamma2)
sol_matrix = np.array([[c * c, c * s_k, s * c, s * s_k, 0., 0.],
[-k * s * c, c * c, -k * s * s, s * c, 0., 0.],
[-s * c, -s * s_k, c * c, c * s_k, 0., 0.],
[k * s * s, -s * c, -k * s * c, c * c, 0., 0.],
[0., 0., 0., 0., 1., r56],
[0., 0., 0., 0., 0., 1.]]).real
return sol_matrix
r_z_e = lambda z, energy: sol(z, k=element.k, energy=energy)
elif element.__class__ == TDCavity:
"""
R - matrix for TDS - NOT TESTED
"""
def tds_R_z(z, energy, freq, v, phi):
"""
:param z: length [m]
:param freq: freq [Hz]
:param v: voltage in [GeV]
:param phi: phase [deg]
:param energy: Energy in [GeV]
:return:
"""
phi = phi * np.pi / 180.
gamma = energy / m_e_GeV
igamma2 = 0.
k0 = 2*np.pi*freq/speed_of_light
if gamma != 0:
igamma2 = 1. / (gamma * gamma)
if gamma > 1:
pref = m_e_GeV * np.sqrt(gamma**2 - 1)
K = v * k0 / pref
else:
K = 0.
cos_phi = np.cos(phi)
cos2_phi = np.cos(2*phi)
rm = np.eye(6)
rm[0, 1] = z
rm[0, 4] = -z * K * cos_phi / 2.
rm[1, 4] = -K * cos_phi
rm[2, 3] = z
rm[4, 5] = - z * igamma2 / (1. - igamma2)
rm[5, 0] = rm[1, 4]
rm[5, 1] = rm[0, 4]
rm[5, 4] = -z* K ** 2 * cos2_phi / 6
return rm
r_z_e = lambda z, energy: tds_R_z(z, energy, freq=element.freq, v=element.v * z / element.l, phi=element.phi)
elif element.__class__ == Matrix:
rm = np.eye(6)
rm = element.r
def r_matrix(z, l, rm):
if z < l:
r_z = uni_matrix(z, 0, hx=0)
else:
r_z = rm
return r_z
r_z_e = lambda z, energy: r_matrix(z, element.l, rm)
elif element.__class__ == Multipole:
r = np.eye(6)
r[1, 0] = -element.kn[1]
r[3, 2] = element.kn[1]
r[1, 5] = element.kn[0]
r_z_e = lambda z, energy: r
elif element.__class__ == XYQuadrupole:
k1 = element.k1
if element.l == 0:
hx = 0.
hy = 0.
else:
hx = k1 * element.x_offs
hy = -k1 * element.y_offs
def r_mtx(z, k1, hx, hy, sum_tilts=0., energy=0.):
# r = element.l/element.angle
# +K - focusing lens , -K - defoc
gamma = energy / m_e_GeV
kx2 = (k1 + hx * hx)
ky2 = hy*hy - k1
kx = np.sqrt(kx2 + 0.j)
ky = np.sqrt(ky2 + 0.j)
cx = np.cos(z * kx).real
cy = np.cos(z * ky).real
sy = (np.sin(ky * z) / ky).real if ky != 0 else z
igamma2 = 0.
if gamma != 0:
igamma2 = 1. / (gamma * gamma)
beta = np.sqrt(1. - igamma2)
if kx != 0:
sx = (np.sin(kx * z) / kx).real
dx = hx / kx2 * (1. - cx)
dy = hy / ky2 * (1. - cy)
r56 = hx * hx * (z - sx) / kx2 / beta ** 2 + hy * hy * (z - sy) / ky2 / beta ** 2
else:
sx = z
dx = z * z * hx / 2.
dy = z * z * hy / 2.
r56 = hx * hx * z ** 3 / 6. / beta ** 2 + hy * hy * z ** 3 / 6. / beta ** 2
r56 -= z / (beta * beta) * igamma2
u_matrix = np.array([[cx, sx, 0., 0., 0., dx / beta],
[-kx2 * sx, cx, 0., 0., 0., sx * hx / beta],
[0., 0., cy, sy, 0., dy / beta],
[0., 0., -ky2 * sy, cy, 0.,sy * hy / beta],
[hx * sx / beta, dx / beta, hy * sy / beta, dy / beta, 1., r56],
[0., 0., 0., 0., 0., 1.]])
if sum_tilts != 0:
u_matrix = np.dot(np.dot(rot_mtx(-sum_tilts), u_matrix), rot_mtx(sum_tilts))
return u_matrix
r_z_e = lambda z, energy: r_mtx(z, k1, hx=hx, hy=hy, sum_tilts=0, energy=energy)
return r_z_e