# Optics calculations

Optics calculations can be performed via the compute module.

## Twiss computation

We can use compute.twiss in order to compute lattice functions as well as phase advance and tunes:

[3]:

import dipas.compute as compute

twiss = compute.twiss(thin)  # Returns a dict.
print(list(twiss), end='\n\n')
print('Q1:', twiss['Q1'], ', Q2:', twiss['Q2'], end='\n\n')
print('Coupling Matrix:', twiss['coupling_matrix'], sep='\n', end='\n\n')
print('One-Turn Matrix:', twiss['one_turn_matrix'], sep='\n', end='\n\n')
print('Lattice:', twiss['lattice'].columns, sep='\n')

['lattice', 'coupling_matrix', 'Q1', 'Q2', 'one_turn_matrix']

Q1: tensor(2.4200) , Q2: tensor(2.4200)

Coupling Matrix:
tensor([[0., 0.],
[0., 0.]])

One-Turn Matrix:
tensor([[-8.7631e-01,  9.2467e-01,  0.0000e+00,  0.0000e+00,  2.7810e-17,
-8.3255e+00],
[-2.5099e-01, -8.7631e-01,  0.0000e+00,  0.0000e+00, -3.4146e-17,
-1.1137e+00],
[ 0.0000e+00,  0.0000e+00, -8.7631e-01,  1.0994e+00,  0.0000e+00,
0.0000e+00],
[ 0.0000e+00,  0.0000e+00, -2.1111e-01, -8.7631e-01,  0.0000e+00,
0.0000e+00],
[ 1.1137e+00,  8.3255e+00,  0.0000e+00,  0.0000e+00,  1.0000e+00,
3.1825e+02],
[ 1.7416e-17,  1.2832e-17,  0.0000e+00,  0.0000e+00,  1.7203e-33,
1.0000e+00]])

Lattice:
Index(['x', 'px', 'y', 'py', 'bx', 'ax', 'mx', 'by', 'ay', 'my', 'dx', 'dpx',
'dy', 'dpy'],
dtype='object')


Here twiss['lattice'] is a pandas.DataFrame with element labels as index and lattice functions as columns.

## Transfer maps

By using compute.transfer_maps we can compute the transfer maps along the lattice. The parameter method let’s us specify how the transfer maps are computed. The following options are available:

• method='local': Compute the local maps of elements, including closed orbit contribution.

• method='accumulate': Compute the cumulative transfer maps w.r.t. to the start of the lattice.

• method='reduce': Compute the combined transfer map for the whole segment.

[4]:

from dipas.compute import transfer_maps

maps = dict(transfer_maps(thin, method='accumulate', labels=True))
print(*maps[lattice[HKicker, 0].label], sep='\n')

tensor([[0.],
[0.],
[0.],
[0.],
[0.],
[0.]])
tensor([[1.0000, 0.8328, 0.0000, 0.0000, 0.0000, 0.0000],
[0.0000, 1.0000, 0.0000, 0.0000, 0.0000, 0.0000],
[0.0000, 0.0000, 1.0000, 0.8328, 0.0000, 0.0000],
[0.0000, 0.0000, 0.0000, 1.0000, 0.0000, 0.0000],
[0.0000, 0.0000, 0.0000, 0.0000, 1.0000, 6.1278],
[0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 1.0000]])
tensor([[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,  -1.2038],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,  -1.2038,   0.0000,   0.0000,   0.0000,   0.0000]],

[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000]],

[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,  -1.2038],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,  -1.2038,   0.0000,   0.0000]],

[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000]],

[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,  -1.2038,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,  -1.2038,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000, -26.5733]],

[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000]]])


Since the kicker represents the full element as a sequence of drift spaces and thin slices, its first order map (i.e. the transfer matrix) is not the identity matrix.

If we want the element local maps instead we can use method='local' (here 0.3755 is the length of the kicker):

[5]:

maps = dict(transfer_maps(thin, method='local', labels=True))
print('Length of Kicker:', lattice[HKicker, 0].l)
print('Transfer Map:', *maps[lattice[HKicker, 0].label], sep='\n')

Length of Kicker: tensor(0.3755)
Transfer Map:
tensor([[0.],
[0.],
[0.],
[0.],
[0.],
[0.]])
tensor([[1.0000, 0.3755, 0.0000, 0.0000, 0.0000, 0.0000],
[0.0000, 1.0000, 0.0000, 0.0000, 0.0000, 0.0000],
[0.0000, 0.0000, 1.0000, 0.3755, 0.0000, 0.0000],
[0.0000, 0.0000, 0.0000, 1.0000, 0.0000, 0.0000],
[0.0000, 0.0000, 0.0000, 0.0000, 1.0000, 2.7629],
[0.0000, 0.0000, 0.0000, 0.0000, 0.0000, 1.0000]])
tensor([[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,  -0.5428],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,  -0.5428,   0.0000,   0.0000,   0.0000,   0.0000]],

[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000]],

[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,  -0.5428],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,  -0.5428,   0.0000,   0.0000]],

[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000]],

[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,  -0.5428,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,  -0.5428,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000, -11.9816]],

[[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   0.0000,   0.0000,   0.0000]]])


## Orbit Response Matrix

Using compute.orm we can compute the orbit response matrix for a given lattice. We need to specify the kickers and monitors to be used, which can be done in a similar way as for selecting lattice elements in general: either we can specify an identifier that selects multiple elements directly, such as a lattice element type or a regex, or we can specify a list of single element identifiers, such as unambiguous labels for example. Let’s compute the horizontal ORM for one of the example lattices:

[6]:

from importlib import resources
from dipas.build import from_file
from dipas.compute import orm
from dipas.elements import HKicker, HMonitor
import dipas.test.sequences

with resources.path(dipas.test.sequences, 'cryring.seq') as path:
lattice = from_file(path)

orm_x, orm_y = orm(lattice, kickers=HKicker, monitors=HMonitor)


Here we don’t need to call makethin beforehand because this will be done inside the orm function. This is necessary because the orm function will temporarily vary the kicker strengths and, as explained above, for each change to the original lattice we need to create a new thin version (i.e. changes to the original lattice are not automatically mapped to any thin versions that have been created before).

[7]:

print('ORM.shape: ', orm_x.shape)
print('Number of HKickers and HMonitors: ', (len(lattice[HKicker]), len(lattice[HMonitor])), end='\n\n')
print('ORM-X\n\n', orm_x, end='\n\n')
print('ORM-Y\n\n', orm_y)

ORM.shape:  torch.Size([9, 12])
Number of HKickers and HMonitors:  (12, 9)

ORM-X

tensor([[ 1.6402,  1.5428,  0.8887,  2.0276,  1.6111,  1.0921, -2.9513,  3.2348,
-2.7059,  1.2091,  2.1282,  2.0422],
[ 0.8140,  1.0098,  1.7728,  0.9366,  1.2233,  1.8082, -1.3954,  0.5891,
0.4745, -1.3684, -0.1676,  0.0053],
[ 0.4583,  0.6885,  1.6933,  1.2629,  0.8969,  1.6400, -0.7518, -0.1673,
1.1597, -1.7372, -0.6953, -0.4921],
[-0.3921, -0.8058, -2.7059, -2.4109, -0.0308,  1.2010,  1.1115,  1.0460,
-2.6066,  3.2348,  1.6820,  1.3166],
[ 1.1925,  1.3288,  1.7242,  0.4483, -1.3899, -2.0740,  1.5688,  1.4899,
-0.4258, -0.7906,  0.5093,  0.6296],
[ 1.4039,  1.7722,  3.2348,  1.8029, -1.3615, -2.7341,  0.9301,  0.8887,
1.0460, -2.6066, -0.4424, -0.1171],
[-1.9135, -2.1046, -2.6066, -0.5499,  2.2668,  3.2892, -2.6391,  1.2091,
0.8887,  1.0460, -0.9556, -1.1243],
[-0.4376, -0.6838, -1.7705, -1.3686,  0.2506,  1.0372,  0.2900, -1.1510,
1.8197,  1.8726,  0.7970,  0.5795],
[ 1.7387,  1.6842,  1.0460, -0.8998, -2.3613, -2.6678,  3.4040, -2.7059,
1.2091,  0.8887,  2.0117,  1.9636]])

ORM-Y

tensor([[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.]])